(* Content-type: application/vnd.wolfram.mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 14.1' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       158,          7]
NotebookDataLength[    266449,       6043]
NotebookOptionsPosition[     88276,       2363]
NotebookOutlinePosition[    251721,       5757]
CellTagsIndexPosition[    251678,       5754]
WindowTitle->4. Prime Factorization | Introduction to Number Theory
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell[TextData[{
 
 CounterBox["Section"],
 StyleBox[" | ", "SectionBar"],
 "Prime Factorization"
}], "Section",
 CounterAssignments->{{"Section", 3}},
 CellID->1383992936,ExpressionUUID->"aba84274-cae8-4d31-a68e-7cd590a35450"],

Cell[TextData[{
 "\[OpenCurlyDoubleQuote]The obvious mathematical breakthrough would be \
development of an easy way to factor large prime numbers.\
\[CloseCurlyDoubleQuote]",
 StyleBox[" ",
  FontSlant->"Italic"],
 StyleBox["\[LongDash] Bill Gates",
  FontSlant->"Plain"]
}], "Text",
 CellChangeTimes->{{3.887868438055534*^9, 3.887868446042699*^9}, {
  3.8888254251411858`*^9, 3.888825428512583*^9}, {3.899861526216876*^9, 
  3.899861539272315*^9}, {3.935482272602641*^9, 3.9354822739497557`*^9}},
 CellID->1980475242,ExpressionUUID->"431f6b64-6b41-48a4-90c8-32f8d3933a9c"],

Cell["\<\
Can you imagine representing integers in a canonical, systematic way? As a \
sum or a product of specific numbers? If we could do it, we then could \
generalize so many things about numbers. We could just examine those little \
bricks that make up the number we want to examine.\
\>", "Text",
 CellChangeTimes->{
  3.89986150235361*^9, {3.8998615952531977`*^9, 3.8998617460937366`*^9}, {
   3.930759202244165*^9, 3.930759202858035*^9}},
 CellID->1223975305,ExpressionUUID->"65f17eeb-da0a-4ea4-9a74-4039423f498e"],

Cell["\<\
Well, it is quite the reality! It turns out, every number can be represented \
as a product of primes. We call it prime factorization.\
\>", "Text",
 CellChangeTimes->{{3.899861753549366*^9, 3.8998617965009108`*^9}, {
  3.930759211644019*^9, 3.9307592121328773`*^9}},
 CellID->983075642,ExpressionUUID->"88ab9ba2-299d-4e8c-99b2-aa54db8127f8"],

Cell[TextData[{
 "The ",
 StyleBox["prime factorization",
  FontSlant->"Italic"],
 " of integers is one of the most important results in number theory. It says \
that every natural number can be written uniquely as a product of primes."
}], "Text",
 CellChangeTimes->{{3.796447658107957*^9, 3.796447675656891*^9}, {
   3.7964477448952513`*^9, 3.796447758997158*^9}, {3.8789879823426123`*^9, 
   3.878987982517415*^9}, {3.8789880558881655`*^9, 3.878988061278989*^9}, {
   3.883641152545209*^9, 3.8836412038865666`*^9}, {3.8836417605222607`*^9, 
   3.8836417617972193`*^9}, {3.8848445904470406`*^9, 3.8848445909284887`*^9}, 
   3.8878746133140554`*^9, {3.9307592165705843`*^9, 3.930759217226795*^9}},
 CellID->1025122044,ExpressionUUID->"6299ef7d-d67f-445a-bfda-6272fb4e7722"],

Cell[TextData[{
 "Much of what follows depends on ",
 StyleBox["unique factorization",
  FontSlant->"Italic"],
 ". It gives an alternative, conceptually simpler way to find a lot of \
numerical characteristics of a number."
}], "Text",
 CellChangeTimes->{{3.796447871690633*^9, 3.7964479080558076`*^9}, {
   3.796448003203313*^9, 3.796448032484544*^9}, 3.7975745013750515`*^9, {
   3.883641261557679*^9, 3.8836412622827625`*^9}, {3.8839994422202735`*^9, 
   3.8839994486275563`*^9}, {3.930759223849971*^9, 3.93075922458565*^9}, {
   3.955696073810054*^9, 3.955696079228764*^9}},
 Background->None,
 CellID->2112377274,ExpressionUUID->"726d9eb2-ba1b-4cb1-aa5a-eb8ce42aeff4"],

Cell["\<\
It turns out, factoring large numbers is a hard task. In fact, secure \
transfers of information (like digital credit card transactions) rely on that \
very fact. Returning to the Bill Gates quote, if we could find an easy way to \
factor large numbers, then the entire system of secure transactions would \
fail.\
\>", "Text",
 CellChangeTimes->{{3.89986180134569*^9, 3.8998618388539333`*^9}, {
  3.8998619061149883`*^9, 3.899861915876313*^9}, {3.899861969064315*^9, 
  3.899862093748775*^9}, {3.930759230099408*^9, 3.930759230434141*^9}, {
  3.943022240614542*^9, 3.9430222412152824`*^9}, {3.949413706645115*^9, 
  3.9494137276702175`*^9}},
 CellID->802375479,ExpressionUUID->"a109a645-464b-4a5c-9f44-7c40c5c69865"],

Cell["\<\
This lesson will be about prime factorization and some things derived from it.\
\>", "Text",
 CellChangeTimes->{{3.8036163161875086`*^9, 3.803616324010045*^9}, {
  3.8036163854998293`*^9, 3.803616412052627*^9}, {3.87898967203932*^9, 
  3.878989700735967*^9}, {3.8836412735769496`*^9, 3.8836412826942573`*^9}, {
  3.8839992208307714`*^9, 3.8839992429697685`*^9}, {3.8839994250978975`*^9, 
  3.883999427693245*^9}, {3.8839994594166155`*^9, 3.8839994655531025`*^9}, {
  3.8893360008958454`*^9, 3.889336001743327*^9}, {3.89986212120674*^9, 
  3.899862121960778*^9}, {3.930759234179285*^9, 3.930759234449086*^9}},
 CellID->525191914,ExpressionUUID->"f760a491-e37d-4f70-a123-445de67eb9b5"],

Cell[CellGroupData[{

Cell["Prime Factorization", "Subsection",
 CellChangeTimes->{{3.7052292751341186`*^9, 3.7052292771423197`*^9}, {
   3.705312795680029*^9, 3.705312804108348*^9}, {3.705336606355554*^9, 
   3.705336630018115*^9}, {3.7054019050053368`*^9, 3.7054019120525713`*^9}, 
   3.705663611118946*^9, {3.708281865271654*^9, 3.708281871750352*^9}, {
   3.708283170503408*^9, 3.70828317254349*^9}, {3.791259689492679*^9, 
   3.7912596909278746`*^9}, {3.7952464398530436`*^9, 3.795246454933439*^9}, {
   3.796200768794861*^9, 3.7962007726358128`*^9}, {3.883999259663508*^9, 
   3.8839992607987537`*^9}},
 CellID->1542400032,ExpressionUUID->"d577cabd-46ec-4803-9d6a-f6b5b53a4df6"],

Cell["Prime numbers are the building blocks of integers.", "Text",
 CellChangeTimes->{{3.7961143986128035`*^9, 3.7961144179381256`*^9}, {
   3.7961174964643464`*^9, 3.79611750191159*^9}, {3.8789880907716455`*^9, 
   3.8789881094332075`*^9}, 3.884844609651441*^9, {3.930759244704788*^9, 
   3.930759244993031*^9}},
 CellID->842234887,ExpressionUUID->"8b97fd45-56d6-469e-a8b6-a0531d89dd78"],

Cell["\<\
Intuitively, that means that every integer is built out of prime numbers. \
\>", "Text",
 CellChangeTimes->{{3.796117510419909*^9, 3.7961175390507507`*^9}, 
   3.7961176084983683`*^9, {3.878989711430144*^9, 3.878989716189756*^9}, {
   3.883641303306669*^9, 3.8836413040704465`*^9}, {3.930759247792877*^9, 
   3.930759247999795*^9}},
 CellID->1377028432,ExpressionUUID->"0183d05e-d071-44b0-9a17-78617ecbe79e"],

Cell["Particularly, every integer is a product of primes.", "Text",
 CellChangeTimes->{{3.796117768525622*^9, 3.7961177913246565`*^9}, 
   3.885371444763217*^9},
 CellID->285592905,ExpressionUUID->"b53a84c6-9628-4ea9-b318-7802593a35f2"],

Cell["Look at these examples:", "Text",
 CellChangeTimes->{{3.7961178877873073`*^9, 3.7961178932442465`*^9}, {
  3.8789897218297243`*^9, 3.8789897222047043`*^9}, {3.880018930630671*^9, 
  3.88001893310993*^9}, {3.943022331378124*^9, 3.943022331537159*^9}},
 CellID->334003290,ExpressionUUID->"0eb20aa9-b09f-40fb-9db8-2ce779570793"],

Cell[BoxData[
 FormBox[
  RowBox[{"6", "=", 
   RowBox[{"2", "*", "3"}]}], TraditionalForm]], "DisplayFormula",
 CellID->57689067,ExpressionUUID->"c7918e49-9270-4708-aefb-7b0dc97241c2"],

Cell[BoxData[
 FormBox[
  RowBox[{"99", "=", 
   RowBox[{"3", "*", "3", "*", "11"}]}], TraditionalForm]], "DisplayFormula",
 CellID->1909381740,ExpressionUUID->"6c6d688f-6804-4142-9b06-8b61337cfec0"],

Cell[BoxData[
 FormBox[
  RowBox[{"600", "=", 
   RowBox[{"2", "*", "2", "*", "2", "*", "3", "*", "5", "*", "5"}]}], 
  TraditionalForm]], "DisplayFormula",
 CellID->1130767059,ExpressionUUID->"72d5dfc5-8e77-429c-abed-375c240c96b4"],

Cell[BoxData[
 FormBox[
  RowBox[{"17", "=", "17"}], TraditionalForm]], "DisplayFormula",
 CellID->1719168310,ExpressionUUID->"59546147-9cf7-4f24-9d6f-cddaec66caf2"],

Cell["\<\
As seen, the primes in the product are neither distinct nor arranged in any \
order; for example,\
\>", "Text",
 CellChangeTimes->{{3.7961182351125965`*^9, 3.796118238797512*^9}, {
  3.7961186029445457`*^9, 3.796118607792718*^9}, {3.7961186549317284`*^9, 
  3.796118686221808*^9}, {3.79620001806293*^9, 3.796200046494235*^9}, {
  3.7962002150997863`*^9, 3.7962002164727244`*^9}, {3.796200263723717*^9, 
  3.796200277575384*^9}, {3.8789610084528713`*^9, 3.8789610541603913`*^9}, {
  3.8789610990740013`*^9, 3.878961108985112*^9}, {3.878989728908876*^9, 
  3.878989730045721*^9}, {3.878989915716732*^9, 3.8789899232892227`*^9}, {
  3.8836413443629475`*^9, 3.883641347717099*^9}, {3.8839986497100873`*^9, 
  3.8839986534696474`*^9}, {3.930759276178369*^9, 3.9307592765901203`*^9}, {
  3.9489034160686455`*^9, 3.9489034376456013`*^9}},
 CellID->214481230,ExpressionUUID->"a7230b2c-3a60-462a-83e6-cca0f2acdd8f"],

Cell[BoxData[
 FormBox[
  RowBox[{"600", "=", 
   RowBox[{"2", "*", "2", "*", "2", "*", "3", "*", "5", "*", "5"}]}], 
  TraditionalForm]], "DisplayFormula",
 CellID->297981691,ExpressionUUID->"40b2cf7d-79bc-40cf-a8ff-f3e4a3051226"],

Cell[BoxData[
 FormBox[
  RowBox[{"600", "=", 
   RowBox[{"2", "*", "3", "*", "2", "*", "5", "*", "2", "*", "5"}]}], 
  TraditionalForm]], "DisplayFormula",
 CellID->246415833,ExpressionUUID->"6088ad3a-4182-4454-9080-2d7981baa62f"],

Cell[BoxData[
 FormBox[
  RowBox[{"600", "=", 
   RowBox[{"3", "*", "5", "*", "2", "*", "2", "*", "2", "*", "5"}]}], 
  TraditionalForm]], "DisplayFormula",
 CellID->1162433815,ExpressionUUID->"e6459dd6-70a4-407c-9115-4e7d239bf9f6"],

Cell["\<\
A nice representation of prime factorization would be achieved by arranging \
primes in increasing order and associating sets of equal primes into single \
factors:\
\>", "Text",
 CellChangeTimes->{{3.796200291153637*^9, 3.7962003139582343`*^9}, {
   3.7962006744555297`*^9, 3.7962007331694202`*^9}, 3.7975755398581176`*^9, {
   3.878961241932583*^9, 3.8789613089304757`*^9}, {3.878989743713174*^9, 
   3.8789897678910275`*^9}, {3.883999319297017*^9, 3.8839993295310774`*^9}, {
   3.884844627472091*^9, 3.884844627736334*^9}, {3.889334912234023*^9, 
   3.8893349125542994`*^9}, {3.930759280679443*^9, 3.9307592813862963`*^9}, {
   3.9430223718012333`*^9, 3.943022376199135*^9}},
 CellID->933304272,ExpressionUUID->"197b7eb2-c365-4c9a-9384-1d9b596769da"],

Cell[BoxData[
 FormBox[
  RowBox[{"600", "=", 
   RowBox[{
    SuperscriptBox["2", "3"], "*", "3", "*", 
    SuperscriptBox["5", "2"]}]}], TraditionalForm]], "DisplayFormula",
 CellID->673185976,ExpressionUUID->"51f8b649-b922-46fa-8737-b68ad040c1c6"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Unique Factorization", "Subsection",
 CellChangeTimes->{{3.7052292751341186`*^9, 3.7052292771423197`*^9}, {
   3.705312795680029*^9, 3.705312804108348*^9}, {3.705312964851019*^9, 
   3.7053129673472967`*^9}, 3.705401949751856*^9, {3.708282353141633*^9, 
   3.7082823565415688`*^9}, {3.791454003414813*^9, 3.791454009220496*^9}, {
   3.7952464660121045`*^9, 3.795246471923724*^9}, {3.7962007783356123`*^9, 
   3.796200780006379*^9}},
 CellID->323327131,ExpressionUUID->"f1a7ee64-bdd9-4600-b39c-f11be83c4c1c"],

Cell["\<\
Up until now, there has been no discussion about the uniqueness of such a \
representation, but a quick examination of special cases immediately \
indicates that this is true.\
\>", "Text",
 CellChangeTimes->{{3.7962016335652657`*^9, 3.796201670817122*^9}, {
  3.796281054890049*^9, 3.796281061518257*^9}, {3.796284392839193*^9, 
  3.7962844200865726`*^9}, {3.8789897831843843`*^9, 3.878989789403255*^9}, {
  3.930759287582921*^9, 3.9307592879694767`*^9}, {3.9430224104252357`*^9, 
  3.9430224106670685`*^9}},
 CellID->1025463807,ExpressionUUID->"5acfcdb1-a2ce-4a8b-9d95-301575c831b3"],

Cell[TextData[{
 "The ",
 StyleBox["fundamental theorem of arithmetic",
  FontSlant->"Italic"],
 " states that every integer greater than 1 can be represented as a product \
of prime factors in only one way, apart from the order of the factors."
}], "Text",
 CellChangeTimes->{{3.796284930094615*^9, 3.796285078542595*^9}, 
   3.7962858555429206`*^9, 3.7962861355044136`*^9, {3.87898981218519*^9, 
   3.878989822409148*^9}, {3.8789900889608016`*^9, 3.8789900929773407`*^9}, {
   3.8789901691040807`*^9, 3.878990179698662*^9}, {3.880019437656417*^9, 
   3.880019438867653*^9}, {3.8836416825765934`*^9, 3.883641713181262*^9}, {
   3.9307592920336246`*^9, 3.9307592921774797`*^9}},
 CellID->1335604159,ExpressionUUID->"bb29dca0-9248-413f-b358-2a4f5600ab74"],

Cell[TextData[{
 "Formally, one can represent the factorization of ",
 Cell[BoxData[
  FormBox["n", TraditionalForm]], "InlineMath",ExpressionUUID->
  "fdf650cc-87b2-4d46-b5f1-a1bd5eb663a7"],
 " as"
}], "Text",
 CellChangeTimes->{{3.7962858646269636`*^9, 3.7962858706681232`*^9}, {
   3.796285921448539*^9, 3.796285929416736*^9}, 3.796286129915594*^9, {
   3.878990208066454*^9, 3.8789902083320627`*^9}, {3.883641727493968*^9, 
   3.883641740156751*^9}, {3.93075930276474*^9, 3.930759307127466*^9}, 
   3.94890346767898*^9},
 CellID->160259201,ExpressionUUID->"d379a323-a521-4726-a598-713c5f78c71a"],

Cell[BoxData[
 FormBox[
  RowBox[{"n", "=", 
   RowBox[{
    SubsuperscriptBox["p", "1", 
     SubscriptBox["k", "1"]], 
    SubsuperscriptBox["p", "2", 
     SubscriptBox["k", "2"]], "\[Ellipsis]", " ", 
    SubsuperscriptBox["p", "m", 
     SubscriptBox["k", "m"]]}]}], TraditionalForm]], "DisplayFormula",
 CellID->98567012,ExpressionUUID->"670e45ea-b165-4c60-860e-2686d8548462"],

Cell[TextData[{
 "where all the ",
 Cell[BoxData[
  FormBox[
   SubscriptBox["p", "i"], TraditionalForm]], "InlineMath",ExpressionUUID->
  "041fca12-f0ca-4dcb-a41a-bbce3e3f1d1e"],
 " are distinct."
}], "Text",
 CellID->780121836,ExpressionUUID->"b5fb5914-dba0-46f9-b12e-c17dbc6357e3"],

Cell[TextData[{
 "Unique factorization is one of the main reasons ",
 Cell[BoxData[
  FormBox["1", TraditionalForm]], "InlineMath",ExpressionUUID->
  "15e322f8-2b9c-4cad-acd2-9375a483513f"],
 " is not considered a prime number. If ",
 Cell[BoxData[
  FormBox["1", TraditionalForm]], "InlineMath",ExpressionUUID->
  "31005bc4-b6b0-614a-b937-1865acab793e"],
 " were considered a prime number, the factorization into primes would no \
longer be unique:"
}], "Text",
 Background->None,
 CellID->1120032717,ExpressionUUID->"6d222f70-ad0f-4df7-8cbb-135e1a6552fa"],

Cell[BoxData[
 FormBox[
  RowBox[{"7", "=", 
   RowBox[{
    RowBox[{"7", "*", "1"}], "=", 
    RowBox[{
     RowBox[{"7", "*", "1", "*", "1"}], "=", "\[Ellipsis]"}]}]}], 
  TraditionalForm]], "DisplayFormula",
 CellID->1285747781,ExpressionUUID->"f408a987-844a-4927-8c16-756c7ec1f651"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Examples", "Subsection",
 CellChangeTimes->{{3.7052292751341186`*^9, 3.7052292771423197`*^9}, {
   3.705312795680029*^9, 3.705312804108348*^9}, {3.705312964851019*^9, 
   3.7053129673472967`*^9}, 3.705401949751856*^9, {3.708282353141633*^9, 
   3.7082823565415688`*^9}, {3.791454003414813*^9, 3.791454009220496*^9}, {
   3.791516138886061*^9, 3.791516148842293*^9}, 3.791516642964593*^9, 
   3.7915171406935215`*^9, {3.7952465426536827`*^9, 3.7952465482205143`*^9}, {
   3.7963599353051767`*^9, 3.7963599472763233`*^9}, {3.889336984793953*^9, 
   3.8893369866728344`*^9}},
 CellID->2142680734,ExpressionUUID->"771370dc-0ad1-4aae-b3ad-32cca02f48e3"],

Cell["Let\[CloseCurlyQuote]s play with the factorization of certain \
integers.", "Text",
 CellChangeTimes->{{3.796360032147696*^9, 3.796360050797017*^9}, {
  3.796361208054245*^9, 3.7963612082361403`*^9}, {3.8789903559820194`*^9, 
  3.878990356466363*^9}, {3.8848446766568165`*^9, 3.884844687694649*^9}},
 CellID->881819795,ExpressionUUID->"7c3161bd-d5f8-4b9e-a9e4-3aa5c564294c"],

Cell["\<\
Obviously, the prime factorization of primes is just themselves because they \
are already prime:\
\>", "Text",
 CellChangeTimes->{{3.7963602259147377`*^9, 3.7963602970459833`*^9}, {
  3.878990371653222*^9, 3.878990373305381*^9}, {3.8893349724514213`*^9, 
  3.8893349727935305`*^9}, {3.9430224719776096`*^9, 3.9430224724911575`*^9}},
 CellID->960646618,ExpressionUUID->"0f572605-edd4-4524-95f6-4ece577bd4f4"],

Cell[BoxData[
 FormBox[
  RowBox[{"23", "=", "23"}], TraditionalForm]], "DisplayFormula",
 CellID->2042058743,ExpressionUUID->"b9dd5c18-4982-4b85-8e0a-958a326a3883"],

Cell[BoxData[
 FormBox[
  RowBox[{"127", "=", "127"}], TraditionalForm]], "DisplayFormula",
 CellID->848621744,ExpressionUUID->"ec3651ce-29f6-4f89-bc32-97c9dc360631"],

Cell["Here are some other examples of prime factorization:", "Text",
 CellChangeTimes->{{3.7963603094828606`*^9, 3.796360321918755*^9}, {
   3.7963604581677074`*^9, 3.79636046374351*^9}, 3.797575560564261*^9, {
   3.803619534539197*^9, 3.803619549872438*^9}, {3.9489036798568935`*^9, 
   3.9489036828716965`*^9}},
 CellID->664558155,ExpressionUUID->"dedf254f-2e43-404c-af40-7a33eb89f6d0"],

Cell[BoxData[
 FormBox[
  RowBox[{"8", "=", 
   SuperscriptBox["2", "3"]}], TraditionalForm]], "DisplayFormula",
 CellID->1776114313,ExpressionUUID->"34ae9ec2-a661-4599-b052-a31cc88124d6"],

Cell[BoxData[
 FormBox[
  RowBox[{"18", "=", 
   RowBox[{"2", "*", 
    SuperscriptBox["3", "2"]}]}], TraditionalForm]], "DisplayFormula",
 CellID->1130565291,ExpressionUUID->"ab8ee790-5d91-4ff7-b443-4f0d19a7b50c"],

Cell[BoxData[
 FormBox[
  RowBox[{"40", "=", 
   RowBox[{
    SuperscriptBox["2", "3"], "*", "5"}]}], 
  TraditionalForm]], "DisplayFormula",
 CellID->475537181,ExpressionUUID->"05a08401-f29e-45fa-ace3-982058d2f900"],

Cell[BoxData[
 FormBox[
  RowBox[{"875", "=", 
   RowBox[{
    SuperscriptBox["5", "3"], "*", "7"}]}], 
  TraditionalForm]], "DisplayFormula",
 CellID->1566750580,ExpressionUUID->"82470124-087e-4e1a-832c-126af6743ca4"],

Cell[BoxData[
 FormBox[
  RowBox[{"1200", "=", 
   RowBox[{
    SuperscriptBox["2", "4"], "*", "3", "*", 
    SuperscriptBox["5", "2"]}]}], TraditionalForm]], "DisplayFormula",
 CellID->2080002193,ExpressionUUID->"78064d91-6a20-4859-820f-03931d7a2134"],

Cell[BoxData[
 FormBox[
  RowBox[{"2310", "=", 
   RowBox[{"2", "*", "3", "*", "5", "*", "7", "*", "11"}]}], 
  TraditionalForm]], "DisplayFormula",
 CellID->1858051885,ExpressionUUID->"15395355-f2ac-40c9-b4b4-bfd5a5dfcf48"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Number of Distinct Prime Factors", "Subsection",
 CellChangeTimes->{{3.8861429578982697`*^9, 3.886142963708105*^9}, {
  3.889337003271962*^9, 3.889337008219509*^9}},
 CellID->1581852389,ExpressionUUID->"a7281f9e-bd38-4e8b-8d2c-5fe6e8792636"],

Cell[TextData[{
 "The plot below shows the number of distinct prime factors of the first ",
 Cell[BoxData[
  FormBox["100", TraditionalForm]], "InlineMath",ExpressionUUID->
  "5bde532e-9431-4746-a8ee-6f7b35f7af51"],
 " numbers:"
}], "Text",
 CellID->345552063,ExpressionUUID->"49f07b95-b228-40a3-85ad-c575e26adc93"],

Cell[BoxData[""], "Input",
 CellChangeTimes->{{3.8893360838252797`*^9, 3.8893360858095465`*^9}},
 CellID->897741353,ExpressionUUID->"21ce2761-a887-44a1-a6fc-031f33432c28"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"DiscretePlot", "[", 
  RowBox[{
   RowBox[{"Length", "[", 
    RowBox[{"FactorInteger", "[", "n", "]"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"n", ",", "100"}], "}"}], ",", 
   RowBox[{"PlotMarkers", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"Automatic", ",", "Small"}], "}"}]}], ",", 
   RowBox[{"AxesOrigin", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"0", ",", "0"}], "}"}]}], ",", 
   RowBox[{"ImageSize", "->", "Medium"}]}], "]"}]], "Input",
 CellID->1607086438,ExpressionUUID->"cc5ff134-f3ff-406b-94a6-ceda0da2a409"],

Cell[BoxData[
 GraphicsBox[{{
    {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.004766666666666667], 
     AbsoluteThickness[2], 
     TagBox[{
       {RGBColor[0.368417, 0.506779, 0.709798], PointSize[
        0.004766666666666667], AbsoluteThickness[2], Opacity[0.2], 
        LineBox[{},
         VertexColors->None]}, 
       {RGBColor[0.368417, 0.506779, 0.709798], PointSize[
        0.004766666666666667], AbsoluteThickness[2], Opacity[0.2], 
        LineBox[CompressedData["
1:eJxt08lKZEEUhOFDu3HpwoULFyoiIiLOs1Z2lfPYXdoOW7HX/QrxaPlIPoJC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         "],
         VertexColors->None]}},
      Annotation[#, "Charting`Private`Tag$78871#1"]& ]}, 
    TagBox[
     {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.004766666666666667],
       AbsoluteThickness[2], {}, GeometricTransformationBox[InsetBox[
        GraphicsBox[
         {EdgeForm[None], DiskBox[{0, 0}]},
         DefaultBaseStyle->{"Graphics", 
           Directive[
            PointSize[0.004766666666666667], 
            RGBColor[0.368417, 0.506779, 0.709798], 
            AbsoluteThickness[2]]},
         PlotRangePadding->Scaled[0.15]], {0, 0}, Automatic, Offset[
        7]], CompressedData["
1:eJxd0btOQkEUheGlNpYWFhYWaowhxhjveAEZBRRvCEfw0hqtfYV5tHkkH0GK
s0z2T3JCvvx7z5zA+udP9b0g6Wv2zM2eefnz24nfStGL8BK8DK+keM4q+hq8
AW9ifwu9gb6NvoO+C+9hfh8+wPwh+hH6MXwCN7F/in6Gfo5+gd5Cb6Nf1vb/
1kFP0dn2eVfRGdY1uu3zu+iweui29/vx/XMf/Qb7sG7Rbe8P0Afod+iw7tFt
7z+gw3pEh/WEbvv8YW3/PkP0Z+zDGsX5NMK8XZ9fMK9xnE9wtr1ve7/CfhV7
Ri+wXrAPZ9v3275/gv1J7Bm9oGsaz0/TOJ/hAusV99u+3/b7Y15v2IczXGC9
Y9/2/egF1se//wD9hoaI
       "]], {}},
     Annotation[#, "Charting`Private`Tag$78871#1"]& ]}, {}, {}},
  AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
  Axes->True,
  AxesOrigin->{0., 0.},
  DisplayFunction->Identity,
  FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
  GridLines->{None, None},
  GridLinesStyle->Directive[
    GrayLevel[0.5, 0.4]],
  ImageSize->Medium,
  Method->{
   "MessagesHead" -> DiscretePlot, "AxisPadding" -> Scaled[0.02], 
    "DefaultBoundaryStyle" -> Automatic, 
    "DefaultGraphicsInteraction" -> {
     "Version" -> 1.2, "TrackMousePosition" -> {True, False}, 
      "Effects" -> {
       "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, 
        "Droplines" -> {
         "freeformCursorMode" -> True, 
          "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" -> 
    AbsolutePointSize[6], "DefaultPlotStyle" -> {
      Directive[
       RGBColor[0.368417, 0.506779, 0.709798], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.880722, 0.611041, 0.142051], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.560181, 0.691569, 0.194885], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.922526, 0.385626, 0.209179], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.528488, 0.470624, 0.701351], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.772079, 0.431554, 0.102387], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.363898, 0.618501, 0.782349], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[1, 0.75, 0], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.647624, 0.37816, 0.614037], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.571589, 0.586483, 0.], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.915, 0.3325, 0.2125], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.40082222609352647`, 0.5220066643438841, 0.85], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.9728288904374106, 0.621644452187053, 0.07336199581899142], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.736782672705901, 0.358, 0.5030266573755369], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.28026441037696703`, 0.715, 0.4292089322474965], 
       AbsoluteThickness[2]]}, "DomainPadding" -> Scaled[0.02], 
    "PointSizeFunction" -> "SmallPointSize", "RangePadding" -> Scaled[0.05]},
  PlotRange->{{0, 100}, {0, 3.}},
  PlotRangeClipping->True,
  PlotRangePadding->{{
     Scaled[0.02], 
     Scaled[0.02]}, {
     Scaled[0.05], 
     Scaled[0.05]}},
  Ticks->{Automatic, Automatic}]], "Output",
 CellChangeTimes->{3.936241648485371*^9},
 CellID->1675591919,ExpressionUUID->"f1c18c41-e7be-44b5-a588-728babf8d392"]
}, {2}]],

Cell[TextData[{
 "That number cannot be greater than three in the first ",
 Cell[BoxData[
  FormBox["100", TraditionalForm]], "InlineMath",ExpressionUUID->
  "2077df7c-ed11-463e-9c83-73f6f10d00cc"],
 " integers because the product of the first four primes is ",
 Cell[BoxData[
  FormBox["210", TraditionalForm]], "InlineMath",ExpressionUUID->
  "318e6574-8097-422b-873b-033886134b3d"],
 " ",
 Cell[BoxData[
  FormBox[
   RowBox[{"(", 
    RowBox[{
     RowBox[{"2", "*", "3", "*", "5", "*", "7"}], "=", "210"}], ")"}], 
   TraditionalForm]], "InlineMath",ExpressionUUID->
  "33f70597-fab8-49b3-bbf7-6672087af9dd"],
 ", which is greater than ",
 Cell[BoxData[
  FormBox["100", TraditionalForm]], "InlineMath",ExpressionUUID->
  "1da52dc6-5de6-4b4d-ade0-286ab4fca53e"],
 "."
}], "Text",
 CellID->604237601,ExpressionUUID->"8c5517ee-d2ed-445c-b693-cb7b157941b6"],

Cell[TextData[{
 "The first number that has three distinct prime factors is ",
 Cell[BoxData[
  FormBox["30", TraditionalForm]], "InlineMath",ExpressionUUID->
  "04f999df-02d7-4323-bdd1-05ffa5bb3d66"],
 " since ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"2", "*", "3", "*", "5"}], "=", "30"}], TraditionalForm]], 
  "InlineMath",ExpressionUUID->"bcaf5a6e-6b00-47f5-8354-7da9488d32d2"],
 "."
}], "Text",
 CellID->1070036956,ExpressionUUID->"3dd53632-f5b9-40f0-b72a-e2dd7f164f9e"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Numerical Characteristics of a Number", "Subsection",
 CellChangeTimes->{{3.7052292751341186`*^9, 3.7052292771423197`*^9}, {
   3.705312795680029*^9, 3.705312804108348*^9}, {3.7053367369530354`*^9, 
   3.705336745435975*^9}, 3.705402169899325*^9, {3.708282403810295*^9, 
   3.708282412710411*^9}, {3.708429197702455*^9, 3.708429202992169*^9}, {
   3.791518577812703*^9, 3.7915185787388906`*^9}, {3.7915186413012667`*^9, 
   3.791518649869476*^9}, {3.7952466081992044`*^9, 3.7952466118311462`*^9}, {
   3.7964399514957266`*^9, 3.7964399557266617`*^9}, {3.886134632997178*^9, 
   3.886134644692628*^9}},
 CellID->399343764,ExpressionUUID->"eb2456e1-dfb2-4faf-a3e8-6356c042df54"],

Cell[TextData[{
 "Have a look at the prime factorization of ",
 Cell[BoxData[
  FormBox["1200", TraditionalForm]], "InlineMath",ExpressionUUID->
  "79809ea2-ad9e-477d-867d-9617b41e883b"],
 ":"
}], "Text",
 CellID->884244673,ExpressionUUID->"acec063d-2d5c-4ae6-8530-8de5a3873249"],

Cell[BoxData[
 FormBox[
  RowBox[{"1200", "=", 
   RowBox[{
    SuperscriptBox["2", "4"], "*", "3", "*", 
    SuperscriptBox["5", "2"]}]}], TraditionalForm]], "DisplayFormula",
 CellID->1426267429,ExpressionUUID->"9b580d62-6053-4424-82af-1b7adc352969"],

Cell["In terms of numbers, it says:", "Text",
 CellChangeTimes->{{3.796390558838972*^9, 3.796390623818591*^9}, {
  3.8836419652211657`*^9, 3.883641967092124*^9}, {3.8848447382166004`*^9, 
  3.884844754110792*^9}, {3.930759355387792*^9, 3.930759355713842*^9}},
 CellID->995997141,ExpressionUUID->"cf5f7ce1-7bd3-42a4-98c4-3d76ada358b4"],

Cell[CellGroupData[{

Cell[TextData[{
 Cell[BoxData[
  FormBox["1200", TraditionalForm]], "InlineMath",ExpressionUUID->
  "71327eb6-e6a3-47d8-ad91-9b311b05551f"],
 " has three distinct prime factors."
}], "Item",
 CellID->88920422,ExpressionUUID->"fc89f60b-c88a-4a0f-b86f-33ebbae16739"],

Cell[TextData[{
 "The exponents of primes in the factorization are ",
 Cell[BoxData[
  FormBox[
   RowBox[{"{", 
    RowBox[{"4", ",", "1", ",", "2"}], "}"}], TraditionalForm]], "InlineMath",
  ExpressionUUID->"6f14e9dd-18b3-4ca4-b8b9-baecd9d0d535"],
 "."
}], "Item",
 CellID->1984397889,ExpressionUUID->"aff0d6d4-5fa1-4004-8d07-75e615dd7afd"],

Cell[TextData[{
 "The sum of the exponents of primes in the factorization is ",
 Cell[BoxData[
  FormBox["7", TraditionalForm]], "InlineMath",ExpressionUUID->
  "73cfbc14-87bb-4066-b317-dbebea23c13a"],
 "."
}], "Item",
 CellID->1354282513,ExpressionUUID->"a4c6fbd6-a2e6-47c2-b8fb-ba2b8b9bac12"]
}, Open  ]],

Cell["\<\
This all is not just for fun. There are very important functions in number \
theory that represent all these characteristics.\
\>", "Text",
 CellChangeTimes->{{3.7963909005453825`*^9, 3.7963909487892995`*^9}, {
  3.8789907691256266`*^9, 3.878990778540762*^9}, {3.930759361703361*^9, 
  3.930759362066464*^9}},
 CellID->247021438,ExpressionUUID->"3eee7785-87bb-4a0b-b19d-d0af07d11ee6"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Liouville Function", "Subsection",
 CellChangeTimes->{{3.7052292751341186`*^9, 3.7052292771423197`*^9}, {
   3.705312795680029*^9, 3.705312804108348*^9}, {3.7053367369530354`*^9, 
   3.705336745435975*^9}, 3.705402169899325*^9, {3.708282403810295*^9, 
   3.708282412710411*^9}, {3.708429197702455*^9, 3.708429202992169*^9}, {
   3.791518577812703*^9, 3.7915185787388906`*^9}, {3.7915186413012667`*^9, 
   3.791518649869476*^9}, {3.7952466081992044`*^9, 3.7952466118311462`*^9}, {
   3.7964399514957266`*^9, 3.7964399557266617`*^9}},
 CellID->1242242571,ExpressionUUID->"5e0ca36d-7a01-4ab7-b299-2df8e409709c"],

Cell[TextData[{
 "The first function of that kind is the",
 StyleBox[" ",
  FontWeight->"Bold"],
 StyleBox["Liouville function ",
  FontSlant->"Italic"],
 Cell[BoxData[
  FormBox["\[Lambda]", TraditionalForm]], "InlineMath",ExpressionUUID->
  "20c6ba97-a8ac-408e-9be0-c481342fdd78"],
 ", which is defined as follows:"
}], "Text",
 CellID->534874663,ExpressionUUID->"b51278b4-83a8-4f6b-b295-15cbf3f8538f"],

Cell[TextData[{
 "If ",
 Cell[BoxData[
  FormBox[
   RowBox[{"n", "=", " ", 
    RowBox[{
     TemplateBox[{"p", "1", 
       SubscriptBox["k", "1"]},
      "Subsuperscript"], " ", 
     TemplateBox[{"p", "2", 
       SubscriptBox["k", "2"]},
      "Subsuperscript"], "\[Ellipsis]", " ", 
     TemplateBox[{"p", "m", 
       SubscriptBox["k", "m"]},
      "Subsuperscript"]}]}], TraditionalForm]], "InlineMath",ExpressionUUID->
  "249c3d9a-dc85-4188-8b25-d7818ddf527f"],
 ", then"
}], "Text",
 CellID->393468604,ExpressionUUID->"307091de-d6dc-41a7-9926-4e14237f06fc"],

Cell[BoxData[
 FormBox[
  RowBox[{
   RowBox[{"\[Lambda]", "(", "n", ")"}], "=", 
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"-", "1"}], ")"}], 
    RowBox[{
     SubscriptBox["k", "1"], "+", 
     RowBox[{"\[Ellipsis]", " ", 
      SubscriptBox["k", "m"]}]}]]}], TraditionalForm]], "DisplayFormula",
 CellID->1744449662,ExpressionUUID->"e55475fa-a625-46dc-886c-982bfde55c17"],

Cell[TextData[{
 "The value of ",
 Cell[BoxData[
  FormBox[
   RowBox[{"\[Lambda]", "(", "1", ")"}], TraditionalForm]], "InlineMath",
  ExpressionUUID->"0f34a91f-6ccd-4b59-9ee9-cd3ff2737281"],
 " is ",
 Cell[BoxData[
  FormBox["1", TraditionalForm]], "InlineMath",ExpressionUUID->
  "8a796ca2-0fc6-485a-b347-a2ec3f41084e"],
 " by definition."
}], "Text",
 CellID->150484525,ExpressionUUID->"25670f9a-5a41-433f-a661-cd227e293494"],

Cell[TextData[{
 "So it only takes two values, ",
 Cell[BoxData[
  FormBox[
   RowBox[{"-", "1"}], TraditionalForm]], "InlineMath",ExpressionUUID->
  "f617ec77-5a21-4ee1-9605-c134643dd7aa"],
 " and ",
 Cell[BoxData[
  FormBox["1", TraditionalForm]], "InlineMath",ExpressionUUID->
  "13930c03-8d15-4c6d-9710-888c7a35f63b"],
 ":"
}], "Text",
 CellID->1605159156,ExpressionUUID->"f9e86950-2a73-4f89-aec7-7ec5a944e9c8"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Grid", "[", 
  RowBox[{
   RowBox[{"Flatten", "/@", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"\"\<\!\(TraditionalForm\`n\)\>\"", ",", 
        RowBox[{"Range", "[", "10", "]"}]}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{"\"\<\!\(TraditionalForm\`\(TraditionalForm\`\[Lambda](n)\)\)\>\
\"", ",", 
        RowBox[{"Table", "[", 
         RowBox[{
          RowBox[{"LiouvilleLambda", "[", "n", "]"}], ",", 
          RowBox[{"{", 
           RowBox[{"n", ",", "10"}], "}"}]}], "]"}]}], "}"}]}], "}"}]}], ",", 
   
   InterpretationBox[
    DynamicModuleBox[{Typeset`open = False}, 
     TemplateBox[{"Expression", "SequenceIcon", 
       GridBox[{{
          RowBox[{
            TagBox["\"Head: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            TagBox["Sequence", "IconizedItem"]}]}, {
          RowBox[{
            TagBox["\"Length: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            TagBox["2", "IconizedItem"]}]}, {
          RowBox[{
            TagBox["\"Byte count: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            
            TagBox["280", "IconizedItem"]}]}}, 
        GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> 
        "Column", 
        GridBoxItemSize -> {
         "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], 
       Dynamic[Typeset`open]},
      "IconizedObject"]],
    Sequence[Frame -> All, Background -> {None, {LightGray, LightBlue}}],
    SelectWithContents->True,
    Selectable->False]}], "]"}]], "Input",
 CellChangeTimes->{{3.7975818482640247`*^9, 3.797581869113061*^9}, 
   3.9362419209561367`*^9},
 CellID->1220361278,ExpressionUUID->"1bb582be-582d-49d1-ab0f-a32b9bd5de10"],

Cell[BoxData[
 TagBox[GridBox[{
    {"\<\"\\!\\(TraditionalForm\\`n\\)\"\>", "1", "2", "3", "4", "5", "6", 
     "7", "8", "9", "10"},
    {"\<\"\\!\\(TraditionalForm\\`\\(TraditionalForm\\`\[Lambda](n)\\)\\)\"\>\
", "1", 
     RowBox[{"-", "1"}], 
     RowBox[{"-", "1"}], "1", 
     RowBox[{"-", "1"}], "1", 
     RowBox[{"-", "1"}], 
     RowBox[{"-", "1"}], "1", "1"}
   },
   AutoDelete->False,
   GridBoxBackground->{"Columns" -> {{None}}, "Rows" -> {
       GrayLevel[0.85], 
       RGBColor[0.87, 0.94, 1]}},
   GridBoxFrame->{"Columns" -> {{True}}, "Rows" -> {{True}}},
   GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}],
  "Grid"]], "Output",
 CellID->858094877,ExpressionUUID->"0e766754-17e7-4cbe-87a3-44088279a363"]
}, {2}]],

Cell[TextData[{
 StyleBox["Here is the plot of the Liouville function for numbers up to ",
  FontWeight->"Plain"],
 Cell[BoxData[
  FormBox["50", TraditionalForm]], "InlineMath",ExpressionUUID->
  "f10a8969-62be-4e5e-8bd1-a8e76be85a97"],
 StyleBox[":",
  FontWeight->"Plain"]
}], "Text",
 CellID->523315141,ExpressionUUID->"74a38599-c47b-466a-8e0a-5f811c92aa02"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"DiscretePlot", "[", 
  RowBox[{
   RowBox[{"LiouvilleLambda", "[", "n", "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"n", ",", "50"}], "}"}], ",", 
   RowBox[{"PlotMarkers", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"Automatic", ",", "Small"}], "}"}]}], ",", 
   RowBox[{"AxesOrigin", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"0", ",", "0"}], "}"}]}], ",", 
   RowBox[{"ImageSize", "->", "Medium"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.936241684716508*^9, 3.9362416869790077`*^9}},
 CellID->1635199970,ExpressionUUID->"996ccd72-3e08-4108-92c6-b271a63cbb05"],

Cell[BoxData[
 GraphicsBox[{{
    {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.009166666666666668], 
     AbsoluteThickness[2], 
     TagBox[{
       {RGBColor[0.368417, 0.506779, 0.709798], PointSize[
        0.009166666666666668], AbsoluteThickness[2], Opacity[0.2], 
        LineBox[CompressedData["
1:eJxdzDsKg1AQheEhaVyA5S1MI2LnAyx8XYLPNksIWGcLs7NsKUtIICrMf2AY
ho85t+frsV1FxP3mss8e/1+ft72PBPAAHsJDuIM7eASP4DE8hifwBJ7CU3gG
z+A5PIcX8AJewkt4Ba/gNbyGN/AG3sJbeAfv4D28h3u4t6536+d9+AAfrMuI
/tG6whUuE/4n6wpX+gyfrcuC/gW+wtfTv2kHT10=
         "],
         VertexColors->None]}, 
       {RGBColor[0.368417, 0.506779, 0.709798], PointSize[
        0.009166666666666668], AbsoluteThickness[2], Opacity[0.2], 
        LineBox[CompressedData["
1:eJxtzD0KhEAMhuGgF7BabBVLERvFsfEXtPYIgrVXyJE9wi7oCHnZD8KQefiS
7Od2hCLy+U3wzJ2r+//6RL3997tPDI/hKTyFZ/AMnsNzeAEv4CW8hFfwCl7D
a3gDb+AO7uAtvLWuvfV39xnQH6wrXOEyoj/CJ/hkXeEKlxn9Gb7AF+u64v76
+heDUzWF
         "],
         VertexColors->None]}},
      Annotation[#, "Charting`Private`Tag$79322#1"]& ]}, 
    TagBox[
     {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.009166666666666668],
       AbsoluteThickness[2], {}, GeometricTransformationBox[InsetBox[
        GraphicsBox[
         {EdgeForm[None], DiskBox[{0, 0}]},
         DefaultBaseStyle->{"Graphics", 
           Directive[
            PointSize[0.009166666666666668], 
            RGBColor[0.368417, 0.506779, 0.709798], 
            AbsoluteThickness[2]]},
         PlotRangePadding->Scaled[0.15]], {0, 0}, Automatic, Offset[
        7]], CompressedData["
1:eJxdzjsKwkAUheGjNpYWFhYWKiIiIhrfj5jxmdi6BMHaLWRnbsklGHAcuH9g
GD7OPXfSfrzuz4qkqDil4pT1/z6JveW837+7Ctecna8jbyBvIm/BHcx34R7m
+/AAHqI/gsfwBP0InsIzeI59C3iJ+RXyNbyBt+jv4BjewwnsrPNg/97BOod1
RB7s95+QwzojD/b9C/4P1hU5rBT7U+QZ+pmd1y3kXwHzZyA=
       "]], {}},
     Annotation[#, "Charting`Private`Tag$79322#1"]& ]}, {}, {}},
  AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
  Axes->True,
  AxesOrigin->{0., 0.},
  DisplayFunction->Identity,
  FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
  GridLines->{None, None},
  GridLinesStyle->Directive[
    GrayLevel[0.5, 0.4]],
  ImageSize->Medium,
  Method->{
   "MessagesHead" -> DiscretePlot, "AxisPadding" -> Scaled[0.02], 
    "DefaultBoundaryStyle" -> Automatic, 
    "DefaultGraphicsInteraction" -> {
     "Version" -> 1.2, "TrackMousePosition" -> {True, False}, 
      "Effects" -> {
       "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, 
        "Droplines" -> {
         "freeformCursorMode" -> True, 
          "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" -> 
    AbsolutePointSize[6], "DefaultPlotStyle" -> {
      Directive[
       RGBColor[0.368417, 0.506779, 0.709798], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.880722, 0.611041, 0.142051], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.560181, 0.691569, 0.194885], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.922526, 0.385626, 0.209179], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.528488, 0.470624, 0.701351], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.772079, 0.431554, 0.102387], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.363898, 0.618501, 0.782349], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[1, 0.75, 0], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.647624, 0.37816, 0.614037], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.571589, 0.586483, 0.], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.915, 0.3325, 0.2125], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.40082222609352647`, 0.5220066643438841, 0.85], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.9728288904374106, 0.621644452187053, 0.07336199581899142], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.736782672705901, 0.358, 0.5030266573755369], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.28026441037696703`, 0.715, 0.4292089322474965], 
       AbsoluteThickness[2]]}, "DomainPadding" -> Scaled[0.02], 
    "PointSizeFunction" -> "SmallPointSize", "RangePadding" -> Scaled[0.05]},
  PlotRange->{{0, 50}, {-1., 1.}},
  PlotRangeClipping->True,
  PlotRangePadding->{{
     Scaled[0.02], 
     Scaled[0.02]}, {
     Scaled[0.05], 
     Scaled[0.05]}},
  Ticks->{Automatic, Automatic}]], "Output",
 CellChangeTimes->{3.936241687535424*^9},
 CellID->1360134510,ExpressionUUID->"df401834-6bc2-44e8-b27a-4e376ba9c6ac"]
}, {2}]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Moebius Function", "Subsection",
 CellChangeTimes->{{3.7052292751341186`*^9, 3.7052292771423197`*^9}, {
   3.705312795680029*^9, 3.705312804108348*^9}, {3.705336758311166*^9, 
   3.705336839987011*^9}, {3.705402179697054*^9, 3.705402188472381*^9}, {
   3.7082824498833313`*^9, 3.7082824545302067`*^9}, {3.710600550944524*^9, 
   3.7106005553286257`*^9}, {3.791518663953273*^9, 3.7915186650233574`*^9}, {
   3.795246619228916*^9, 3.795246622034294*^9}, {3.796439959121807*^9, 
   3.7964399752017565`*^9}, 3.8800188382964563`*^9, {3.8861430372477226`*^9, 
   3.886143037785206*^9}},
 CellID->981804752,ExpressionUUID->"c9c90112-24b7-482d-b174-3171aceb1d5a"],

Cell[TextData[{
 "The next very important function is the",
 StyleBox[" ",
  FontWeight->"Bold"],
 StyleBox["Moebius function ",
  FontSlant->"Italic"],
 Cell[BoxData[
  FormBox["\[Mu]", TraditionalForm]], "InlineMath",ExpressionUUID->
  "08b5d184-71aa-4889-9b10-5d052a5af0dc"],
 ", which is defined as follows:"
}], "Text",
 CellID->2075606963,ExpressionUUID->"041886e3-cfc6-410e-8c92-1a2f291b42fb"],

Cell[TextData[{
 "If ",
 Cell[BoxData[
  FormBox[
   RowBox[{"n", ">", "1"}], TraditionalForm]], "InlineMath",ExpressionUUID->
  "a686a4b9-cb22-4147-a60c-87915dc4b6e5"],
 ", write ",
 Cell[BoxData[
  FormBox[
   RowBox[{"n", "=", " ", 
    RowBox[{
     TemplateBox[{"p", "1", 
       SubscriptBox["k", "1"]},
      "Subsuperscript"], " ", 
     TemplateBox[{"p", "2", 
       SubscriptBox["k", "2"]},
      "Subsuperscript"], " ", 
     TemplateBox[{
       RowBox[{"\[Ellipsis]", " ", "p"}], "m", 
       SubscriptBox["k", "m"]},
      "Subsuperscript"]}]}], TraditionalForm]], "InlineMath",ExpressionUUID->
  "f9f6c3e4-9626-4b30-8405-b2e0631dfaa6"],
 ". Then"
}], "Text",
 CellID->409894930,ExpressionUUID->"d6508ebc-d568-4a87-8daf-ae7bab118256"],

Cell[BoxData[
 FormBox[
  RowBox[{
   RowBox[{
    RowBox[{"\[Mu]", "(", "n", ")"}], "=", 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"-", "1"}], ")"}], "m"]}], ",", " ", 
   RowBox[{
    RowBox[{"if", " ", 
     SubscriptBox["k", "1"]}], "=", 
    RowBox[{
     SubscriptBox["k", "2"], "=", 
     RowBox[{"\[Ellipsis]", "=", 
      RowBox[{
       SubscriptBox["k", "m"], "=", "1"}]}]}]}]}], 
  TraditionalForm]], "DisplayFormula",
 CellID->1730623447,ExpressionUUID->"4955f0ba-2a6d-48ea-b598-fb1fd4ef6d46"],

Cell[BoxData[
 FormBox[
  RowBox[{
   RowBox[{"\[Mu]", "(", "n", ")"}], "=", 
   RowBox[{"0", " ", "otherwise"}]}], TraditionalForm]], "DisplayFormula",
 CellID->1590926560,ExpressionUUID->"8dcd7fcf-6c7d-4163-b04d-62df3017ac07"],

Cell[TextData[{
 "The value of ",
 Cell[BoxData[
  FormBox[
   RowBox[{"\[Mu]", "(", "1", ")"}], TraditionalForm]], "InlineMath",
  ExpressionUUID->"059237de-9a73-4765-a2c3-4042981d83eb"],
 " is ",
 Cell[BoxData[
  FormBox["1", TraditionalForm]], "InlineMath",ExpressionUUID->
  "77e9a82e-b6ec-4cdf-8009-d2dd8fdb46de"],
 " by definition."
}], "Text",
 CellID->59849103,ExpressionUUID->"5a5acc43-22fa-4a3d-8db6-c59b791783d8"],

Cell[TextData[{
 "So it takes three values, ",
 Cell[BoxData[
  FormBox[
   RowBox[{"-", "1"}], TraditionalForm]], "InlineMath",ExpressionUUID->
  "a89584db-2288-4281-9cba-f9213a83ad58"],
 ", ",
 Cell[BoxData[
  FormBox["0", TraditionalForm]], "InlineMath",ExpressionUUID->
  "f29f0d57-4151-47fe-8451-d7289edc9755"],
 " and ",
 Cell[BoxData[
  FormBox["1", TraditionalForm]], "InlineMath",ExpressionUUID->
  "8c4c9ab7-c8d4-4b09-bdd5-25ff06fffdcc"],
 ":"
}], "Text",
 CellID->861527325,ExpressionUUID->"5841051a-73aa-43b3-bf34-56cf38406680"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Grid", "[", 
  RowBox[{
   RowBox[{"Flatten", "/@", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"\"\<\!\(TraditionalForm\`n\)\>\"", ",", 
        RowBox[{"Range", "[", "10", "]"}]}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{"\"\<\!\(\*FormBox[FormBox[FormBox[GridBox[{{RowBox[{\"\[Mu]\",\
 \"(\", \"n\", \")\"}]}
},GridBoxAlignment->{\"Columns\" -> \
{{\"=\"}}}],TraditionalForm],TraditionalForm],TraditionalForm]\)\>\"", ",", 
        RowBox[{"Table", "[", 
         RowBox[{
          RowBox[{"MoebiusMu", "[", "n", "]"}], ",", 
          RowBox[{"{", 
           RowBox[{"n", ",", "10"}], "}"}]}], "]"}]}], "}"}]}], "}"}]}], ",", 
   
   InterpretationBox[
    DynamicModuleBox[{Typeset`open = False}, 
     TemplateBox[{"Expression", "SequenceIcon", 
       GridBox[{{
          RowBox[{
            TagBox["\"Head: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            TagBox["Sequence", "IconizedItem"]}]}, {
          RowBox[{
            TagBox["\"Length: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            TagBox["2", "IconizedItem"]}]}, {
          RowBox[{
            TagBox["\"Byte count: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            
            TagBox["280", "IconizedItem"]}]}}, 
        GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> 
        "Column", 
        GridBoxItemSize -> {
         "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], 
       Dynamic[Typeset`open]},
      "IconizedObject"]],
    Sequence[Frame -> All, Background -> {None, {LightGray, LightBlue}}],
    SelectWithContents->True,
    Selectable->False]}], "]"}]], "Input",
 CellChangeTimes->{{3.797581904490791*^9, 3.797581926387265*^9}, 
   3.936241952013123*^9},
 CellID->1355003201,ExpressionUUID->"0d42e194-b9a8-47ff-8510-6070e0dc6b9b"],

Cell[BoxData[
 TagBox[GridBox[{
    {"\<\"\\!\\(TraditionalForm\\`n\\)\"\>", "1", "2", "3", "4", "5", "6", 
     "7", "8", "9", "10"},
    {"\<\"\\!\\(\\*FormBox[FormBox[FormBox[GridBox[{{RowBox[{\\\"\[Mu]\\\", \
\\\"(\\\", \\\"n\\\", \\\")\\\"}]}\\n    },GridBoxAlignment->{\\\"Columns\\\" \
-> {{\\\"=\\\"}}}],TraditionalForm],TraditionalForm],TraditionalForm]\\)\"\>",
      "1", 
     RowBox[{"-", "1"}], 
     RowBox[{"-", "1"}], "0", 
     RowBox[{"-", "1"}], "1", 
     RowBox[{"-", "1"}], "0", "0", "1"}
   },
   AutoDelete->False,
   GridBoxBackground->{"Columns" -> {{None}}, "Rows" -> {
       GrayLevel[0.85], 
       RGBColor[0.87, 0.94, 1]}},
   GridBoxFrame->{"Columns" -> {{True}}, "Rows" -> {{True}}},
   GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}],
  "Grid"]], "Output",
 CellChangeTimes->{3.7975819273936553`*^9, 3.936241953238214*^9},
 CellID->56005488,ExpressionUUID->"c7c99c40-4ae7-4612-8e9e-cf6e5a71b79d"]
}, {2}]],

Cell[TextData[{
 StyleBox["Here is the plot of the Moebius function for numbers up to ",
  FontWeight->"Plain"],
 Cell[BoxData[
  FormBox["50", TraditionalForm]], "InlineMath",ExpressionUUID->
  "478cab3b-cb73-46a3-8309-9f10a001b3f6"],
 StyleBox[":",
  FontWeight->"Plain"]
}], "Text",
 CellID->2118385468,ExpressionUUID->"5c778cc6-a4b3-462f-88ba-ccf4b81b1939"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"DiscretePlot", "[", 
  RowBox[{
   RowBox[{"MoebiusMu", "[", "n", "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"n", ",", "50"}], "}"}], ",", 
   RowBox[{"PlotMarkers", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"Automatic", ",", "Small"}], "}"}]}], ",", 
   RowBox[{"AxesOrigin", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"0", ",", "0"}], "}"}]}], ",", 
   RowBox[{"ImageSize", "->", "Medium"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.936241714849626*^9, 3.936241716715362*^9}},
 CellID->72970917,ExpressionUUID->"bcc4f2b4-ff43-4ed2-a617-ea56101d308b"],

Cell[BoxData[
 GraphicsBox[{{
    {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.009166666666666668], 
     AbsoluteThickness[2], 
     TagBox[{
       {RGBColor[0.368417, 0.506779, 0.709798], PointSize[
        0.009166666666666668], AbsoluteThickness[2], Opacity[0.2], 
        LineBox[CompressedData["
1:eJyFzj0KwjAAhuGgS0eHjh1UpEgna4Uq2h9Era4eQXD2CrmZV/IICjZCXjBf
IZSHt1/o5Pa43ofGmNnnDPrTP+339Xr6dk+EHqGPhGPsY/QEPUEfC0+FU9yf
os+FM+wz9IVwjn2OvhQusC/QV8Il9iX6WngjvBXeCVf4vwq9Rq/RG/QGvRXe
h22d+/stujmEbZ3dnt8f/f6z26NbdHMK2yp3uL/zuzmHbYXN5a/fFPRFIg==

         "],
         VertexColors->None]}, 
       {RGBColor[0.368417, 0.506779, 0.709798], PointSize[
        0.009166666666666668], AbsoluteThickness[2], Opacity[0.2], 
        LineBox[CompressedData["
1:eJxtyzEKgDAMheGgu7Or4Fi6tUMXFYfuHkFw9go5skdQ0Bby44MQko837Od2
tCLSPdN88+aa/ndJP9t/uUtG+Ah3cAf3cA8P8ACP8AhP8GRdZ+v1LlnQX6wr
XOGyor9aV7jCJaOfq99wcCPD
         "],
         VertexColors->None]}},
      Annotation[#, "Charting`Private`Tag$79348#1"]& ]}, 
    TagBox[
     {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.009166666666666668],
       AbsoluteThickness[2], {}, GeometricTransformationBox[InsetBox[
        GraphicsBox[
         {EdgeForm[None], DiskBox[{0, 0}]},
         DefaultBaseStyle->{"Graphics", 
           Directive[
            PointSize[0.009166666666666668], 
            RGBColor[0.368417, 0.506779, 0.709798], 
            AbsoluteThickness[2]]},
         PlotRangePadding->Scaled[0.15]], {0, 0}, Automatic, Offset[
        7]], CompressedData["
1:eJxdzTsKAjEURuFfbSwtLCwsVERERHR8Pye+tXUJgrVbyM7ckktwwNyBnIEQ
Pu6cm+bz/XiVJCXZKWSnKPu+aXzLBX/+dxmuOEVfFfOaOeyrY95A34Lb6Dvo
u/i/h3kf/QAeoh+hTzAfYz7BfIr9M3iOfoF+Ca/QrzHfwFvs38Ep7OLem+29
fWwP64DebPuP6GGd0JutP8f2sC7oYV3xfu7Q39DDuuf+AU7cSyM=
       "]], {}},
     Annotation[#, "Charting`Private`Tag$79348#1"]& ]}, {}, {}},
  AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
  Axes->True,
  AxesOrigin->{0., 0.},
  DisplayFunction->Identity,
  FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
  GridLines->{None, None},
  GridLinesStyle->Directive[
    GrayLevel[0.5, 0.4]],
  ImageSize->Medium,
  Method->{
   "MessagesHead" -> DiscretePlot, "AxisPadding" -> Scaled[0.02], 
    "DefaultBoundaryStyle" -> Automatic, 
    "DefaultGraphicsInteraction" -> {
     "Version" -> 1.2, "TrackMousePosition" -> {True, False}, 
      "Effects" -> {
       "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, 
        "Droplines" -> {
         "freeformCursorMode" -> True, 
          "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" -> 
    AbsolutePointSize[6], "DefaultPlotStyle" -> {
      Directive[
       RGBColor[0.368417, 0.506779, 0.709798], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.880722, 0.611041, 0.142051], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.560181, 0.691569, 0.194885], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.922526, 0.385626, 0.209179], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.528488, 0.470624, 0.701351], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.772079, 0.431554, 0.102387], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.363898, 0.618501, 0.782349], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[1, 0.75, 0], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.647624, 0.37816, 0.614037], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.571589, 0.586483, 0.], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.915, 0.3325, 0.2125], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.40082222609352647`, 0.5220066643438841, 0.85], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.9728288904374106, 0.621644452187053, 0.07336199581899142], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.736782672705901, 0.358, 0.5030266573755369], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.28026441037696703`, 0.715, 0.4292089322474965], 
       AbsoluteThickness[2]]}, "DomainPadding" -> Scaled[0.02], 
    "PointSizeFunction" -> "SmallPointSize", "RangePadding" -> Scaled[0.05]},
  PlotRange->{{0, 50}, {-1., 1.}},
  PlotRangeClipping->True,
  PlotRangePadding->{{
     Scaled[0.02], 
     Scaled[0.02]}, {
     Scaled[0.05], 
     Scaled[0.05]}},
  Ticks->{Automatic, Automatic}]], "Output",
 CellChangeTimes->{3.936241717221162*^9},
 CellID->1850105899,ExpressionUUID->"eb54da94-c688-4de6-a349-8173000cafb1"]
}, {2}]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Mangoldt\[CloseCurlyQuote]s Function", "Subsection",
 CellChangeTimes->{{3.7052292751341186`*^9, 3.7052292771423197`*^9}, {
   3.705312795680029*^9, 3.705312804108348*^9}, {3.705336758311166*^9, 
   3.705336839987011*^9}, {3.705402179697054*^9, 3.705402188472381*^9}, {
   3.7082824498833313`*^9, 3.7082824545302067`*^9}, {3.710600550944524*^9, 
   3.7106005553286257`*^9}, {3.791518663953273*^9, 3.7915186650233574`*^9}, {
   3.795246619228916*^9, 3.795246622034294*^9}, {3.796439959121807*^9, 
   3.7964399752017565`*^9}, 3.8800188382964563`*^9, 3.886134711995474*^9, 
   3.8893374452035885`*^9},
 CellID->779360166,ExpressionUUID->"369b4c3e-184a-4c97-9079-1791d506fc7d"],

Cell[TextData[{
 "The next function is ",
 StyleBox["Mangoldt\[CloseCurlyQuote]s function ",
  FontSlant->"Italic"],
 Cell[BoxData[
  FormBox["\[CapitalLambda]", TraditionalForm]], "InlineMath",ExpressionUUID->
  "0ab32efd-6e9a-4146-8f1b-a559c6539da2"],
 StyleBox[". ",
  FontWeight->"Plain"],
 "It plays a central role in the distribution of primes. It is defined as \
follows: "
}], "Text",
 CellID->1203546909,ExpressionUUID->"abcb5752-d0dc-4e88-aa14-b1ad0a3a7693"],

Cell[BoxData[
 FormBox[
  RowBox[{
   RowBox[{
    RowBox[{"\[CapitalLambda]", "(", "n", ")"}], "=", 
    RowBox[{"log", "(", "p", ")"}]}], ",", " ", 
   RowBox[{
    RowBox[{"if", " ", "n"}], "=", 
    RowBox[{
     RowBox[{
      SuperscriptBox["p", 
       RowBox[{"m", " "}]], "for", " ", "some", " ", "prime", " ", "p", " ", 
      "and", " ", "m"}], "\[GreaterEqual]", "1"}]}]}], 
  TraditionalForm]], "DisplayFormula",
 CellID->808379129,ExpressionUUID->"8a5ebb1c-d5cc-4485-a8d7-1efb34698476"],

Cell[BoxData[
 FormBox[
  RowBox[{
   RowBox[{"\[CapitalLambda]", "(", "n", ")"}], "=", 
   RowBox[{"0", " ", "otherwise"}]}], TraditionalForm]], "DisplayFormula",
 CellID->1446470181,ExpressionUUID->"c169dbe6-ce67-4c2e-a44c-cc411f5b0b77"],

Cell[TextData[{
 "Here is a short table of values of ",
 Cell[BoxData[
  FormBox[
   RowBox[{"\[CapitalLambda]", "(", "n", ")"}], TraditionalForm]], 
  "InlineMath",ExpressionUUID->"9e2c719a-f017-4f0b-a444-ddb2d2cbb37f"],
 ":"
}], "Text",
 CellID->722423335,ExpressionUUID->"dc5ab057-6391-43e7-bb64-57a3de41dee9"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Grid", "[", 
  RowBox[{
   RowBox[{"Flatten", "/@", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"\"\<\!\(TraditionalForm\`n\)\>\"", ",", 
        RowBox[{"Range", "[", "10", "]"}]}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{"\"\<\!\(\*FormBox[FormBox[FormBox[GridBox[{{FormBox[RowBox[{\"\
\[CapitalLambda]\", \"(\", \"n\", \")\"}],TraditionalForm]}
},GridBoxAlignment->{\"Columns\" -> \
{{\"=\"}}}],TraditionalForm],TraditionalForm],TraditionalForm]\)\>\"", ",", 
        RowBox[{"Table", "[", 
         RowBox[{
          RowBox[{
           RowBox[{"MangoldtLambda", "[", "n", "]"}], "//", 
           "TraditionalForm"}], ",", 
          RowBox[{"{", 
           RowBox[{"n", ",", "10"}], "}"}]}], "]"}]}], "}"}]}], "}"}]}], ",", 
   
   InterpretationBox[
    DynamicModuleBox[{Typeset`open = False}, 
     TemplateBox[{"Expression", "SequenceIcon", 
       GridBox[{{
          RowBox[{
            TagBox["\"Head: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            TagBox["Sequence", "IconizedItem"]}]}, {
          RowBox[{
            TagBox["\"Length: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            TagBox["2", "IconizedItem"]}]}, {
          RowBox[{
            TagBox["\"Byte count: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            
            TagBox["280", "IconizedItem"]}]}}, 
        GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> 
        "Column", 
        GridBoxItemSize -> {
         "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], 
       Dynamic[Typeset`open]},
      "IconizedObject"]],
    Sequence[Frame -> All, Background -> {None, {LightGray, LightBlue}}],
    SelectWithContents->True,
    Selectable->False]}], "]"}]], "Input",
 CellChangeTimes->{{3.7975819641186237`*^9, 3.7975819763006425`*^9}, 
   3.936242028336293*^9},
 CellID->2043934854,ExpressionUUID->"016e8c2e-64cd-4d75-b433-a7946758b1cb"],

Cell[BoxData[
 TagBox[GridBox[{
    {"\<\"\\!\\(TraditionalForm\\`n\\)\"\>", "1", "2", "3", "4", "5", "6", 
     "7", "8", "9", "10"},
    {"\<\"\\!\\(\\*FormBox[FormBox[FormBox[GridBox[{{FormBox[RowBox[{\\\"\
\[CapitalLambda]\\\", \\\"(\\\", \\\"n\\\", \\\")\\\"}],TraditionalForm]}\\n  \
  },GridBoxAlignment->{\\\"Columns\\\" -> \
{{\\\"=\\\"}}}],TraditionalForm],TraditionalForm],TraditionalForm]\\)\"\>", 
     TagBox[
      FormBox["0",
       TraditionalForm],
      TraditionalForm,
      Editable->True], 
     TagBox[
      FormBox[
       RowBox[{"log", "(", "2", ")"}],
       TraditionalForm],
      TraditionalForm,
      Editable->True], 
     TagBox[
      FormBox[
       RowBox[{"log", "(", "3", ")"}],
       TraditionalForm],
      TraditionalForm,
      Editable->True], 
     TagBox[
      FormBox[
       RowBox[{"log", "(", "2", ")"}],
       TraditionalForm],
      TraditionalForm,
      Editable->True], 
     TagBox[
      FormBox[
       RowBox[{"log", "(", "5", ")"}],
       TraditionalForm],
      TraditionalForm,
      Editable->True], 
     TagBox[
      FormBox["0",
       TraditionalForm],
      TraditionalForm,
      Editable->True], 
     TagBox[
      FormBox[
       RowBox[{"log", "(", "7", ")"}],
       TraditionalForm],
      TraditionalForm,
      Editable->True], 
     TagBox[
      FormBox[
       RowBox[{"log", "(", "2", ")"}],
       TraditionalForm],
      TraditionalForm,
      Editable->True], 
     TagBox[
      FormBox[
       RowBox[{"log", "(", "3", ")"}],
       TraditionalForm],
      TraditionalForm,
      Editable->True], 
     TagBox[
      FormBox["0",
       TraditionalForm],
      TraditionalForm,
      Editable->True]}
   },
   AutoDelete->False,
   GridBoxBackground->{"Columns" -> {{None}}, "Rows" -> {
       GrayLevel[0.85], 
       RGBColor[0.87, 0.94, 1]}},
   GridBoxFrame->{"Columns" -> {{True}}, "Rows" -> {{True}}},
   GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}],
  "Grid"]], "Output",
 CellChangeTimes->{3.936242036169723*^9},
 CellID->1450561172,ExpressionUUID->"a6837e00-3b12-41db-9ed1-7ef1a3df480f"]
}, {2}]],

Cell[TextData[{
 StyleBox["Here is the plot of Mangoldt\[CloseCurlyQuote]s function for \
numbers up to ",
  FontWeight->"Plain"],
 Cell[BoxData[
  FormBox["50", TraditionalForm]], "InlineMath",ExpressionUUID->
  "69b216b2-1c15-40bb-b76a-683e893d1dcb"],
 ":"
}], "Text",
 CellID->1726330803,ExpressionUUID->"d36ae15b-d1a2-4743-8750-ff89c44750d0"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"DiscretePlot", "[", 
  RowBox[{
   RowBox[{"MangoldtLambda", "[", "n", "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"n", ",", "50"}], "}"}], ",", 
   RowBox[{"PlotMarkers", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"Automatic", ",", "Small"}], "}"}]}], ",", 
   RowBox[{"AxesOrigin", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"0", ",", "0"}], "}"}]}], ",", 
   RowBox[{"ImageSize", "->", "Medium"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.93624173377984*^9, 3.936241735191481*^9}, {
  3.936242200830315*^9, 3.936242207241889*^9}},
 CellID->1568066294,ExpressionUUID->"5082b51c-2bd9-4829-a276-33d3ad2cd471"],

Cell[BoxData[
 GraphicsBox[{{
    {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.009166666666666668], 
     AbsoluteThickness[2], 
     TagBox[{
       {RGBColor[0.368417, 0.506779, 0.709798], PointSize[
        0.009166666666666668], AbsoluteThickness[2], Opacity[0.2], 
        LineBox[CompressedData["
1:eJyFxzsKg1AYBtEP3YRgFbBIIRYhikjwFY2xdQmCtVu4S84SIpgmgv8MTHEu
8zotvqRw2/u996n019FBbTsCX8ExOAHfwHdwCs7AObgAP8ClbVfbVmPbgdWC
n7YdWB24B79sO7AG8Bs8nvoLweIhDg==
         "],
         VertexColors->None]}, 
       {RGBColor[0.368417, 0.506779, 0.709798], PointSize[
        0.009166666666666668], AbsoluteThickness[2], Opacity[0.2], 
        LineBox[CompressedData["
1:eJxTTMoPSmVmYGAQB2ImKIYCh/eWv/456T2zh/EZUACHAzfz2qpXkz/aw/io
8gJo+gXQ5EUcjHvffnc88NMexkeVl3EIjzJaV6T03x7GR5VXQDNfAU1eCc19
Smjyag4+5vXbH+sxO8D4qPJaDoE9htosrSwOMD6qvAGa/QZo8oYOVfW/Dmav
YnOA8VHljR10NrOwmfWxO8D4qPLmDh82X/xbK8rpAOOjyluihZ8lmrw1mv+t
0eRtHeT/nT1t8YHLAcZHlbd3iNKe6HGugtsBxkeVd0DzvwOqfIOTw97jVu80
HvM4wPio8i4Ofmv6XFO38jrA+Kjyrg6Cb0TEPojxOcD4qPLuDts+TNCXOguT
d0eT90BNP0A+TAoA+9lyGQ==
         "],
         VertexColors->None]}},
      Annotation[#, "Charting`Private`Tag$80909#1"]& ]}, 
    TagBox[
     {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.009166666666666668],
       AbsoluteThickness[2], {}, GeometricTransformationBox[InsetBox[
        GraphicsBox[
         {EdgeForm[None], DiskBox[{0, 0}]},
         DefaultBaseStyle->{"Graphics", 
           Directive[
            PointSize[0.009166666666666668], 
            RGBColor[0.368417, 0.506779, 0.709798], 
            AbsoluteThickness[2]]},
         PlotRangePadding->Scaled[0.15]], {0, 0}, Automatic, Offset[
        7]], CompressedData["
1:eJxTTMoPSmVmYGAwAmJGIGZigIEP9gyowOG95a9/TnrPoOIcDtzMa6teTf4I
5QugyYs4GPe+/e544CeUL+GAapyMQ3iU0boipf9QeQU0/Upo5qug6Vdz8DGv
3/5YjxkqroEmr+UQ2GOozdLKAhXXQZPXQ+MboNlv6FBV/+tg9io2qDojNPXG
DjqbWdjM+tih4iZo8qZofDM0vrnDh80X/9aKckLFLdDkLdHCzwpN3hotfGzQ
5G0d5P+dPW3xgQsqbocmb+8QpT3R41wFN1TcAdX/DQ5o6h1R+Q1ofAYnNHkn
h73Hrd5pPOaBijujyaPxGVzQ5F0c/Nb0uaZu5YWKu6LJuzoIvhER+yDGBxV3
Q5NH4zO4o8m7O2z7MEFf6ixMvweavAda+vSEywMAyYF1ZQ==
       "]], {}},
     Annotation[#, "Charting`Private`Tag$80909#1"]& ]}, {}, {}},
  AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
  Axes->True,
  AxesOrigin->{0., 0.},
  DisplayFunction->Identity,
  FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
  GridLines->{None, None},
  GridLinesStyle->Directive[
    GrayLevel[0.5, 0.4]],
  ImageSize->Medium,
  Method->{
   "MessagesHead" -> DiscretePlot, "AxisPadding" -> Scaled[0.02], 
    "DefaultBoundaryStyle" -> Automatic, 
    "DefaultGraphicsInteraction" -> {
     "Version" -> 1.2, "TrackMousePosition" -> {True, False}, 
      "Effects" -> {
       "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, 
        "Droplines" -> {
         "freeformCursorMode" -> True, 
          "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" -> 
    AbsolutePointSize[6], "DefaultPlotStyle" -> {
      Directive[
       RGBColor[0.368417, 0.506779, 0.709798], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.880722, 0.611041, 0.142051], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.560181, 0.691569, 0.194885], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.922526, 0.385626, 0.209179], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.528488, 0.470624, 0.701351], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.772079, 0.431554, 0.102387], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.363898, 0.618501, 0.782349], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[1, 0.75, 0], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.647624, 0.37816, 0.614037], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.571589, 0.586483, 0.], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.915, 0.3325, 0.2125], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.40082222609352647`, 0.5220066643438841, 0.85], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.9728288904374106, 0.621644452187053, 0.07336199581899142], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.736782672705901, 0.358, 0.5030266573755369], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.28026441037696703`, 0.715, 0.4292089322474965], 
       AbsoluteThickness[2]]}, "DomainPadding" -> Scaled[0.02], 
    "PointSizeFunction" -> "SmallPointSize", "RangePadding" -> Scaled[0.05]},
  PlotRange->{{0, 50}, {0, 3.8501476017100584`}},
  PlotRangeClipping->True,
  PlotRangePadding->{{
     Scaled[0.02], 
     Scaled[0.02]}, {
     Scaled[0.05], 
     Scaled[0.05]}},
  Ticks->{Automatic, Automatic}]], "Output",
 CellChangeTimes->{
  3.9362417356382523`*^9, {3.936242201658992*^9, 3.936242207819536*^9}},
 CellID->826107982,ExpressionUUID->"fe2a8d8c-2873-4502-9dbc-ab273cf584c7"]
}, {2}]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Prime Omega Functions ", "Subsection",
 CellChangeTimes->{{3.7929945775465393`*^9, 3.7929945894597263`*^9}, {
  3.7933345298281517`*^9, 3.7933345442659006`*^9}, {3.79524658291366*^9, 
  3.7952465854312215`*^9}, {3.796438341389743*^9, 3.7964383436971626`*^9}, {
  3.7964386574323964`*^9, 3.796438665638298*^9}, {3.8893373458434944`*^9, 
  3.8893373463425655`*^9}, {3.943022957857609*^9, 3.943022982851164*^9}},
 CellID->1158490424,ExpressionUUID->"846307f5-2eaf-4f36-94ae-5e64be2f4627"],

Cell[TextData[{
 "The ",
 StyleBox["prime omega functions",
  FontSlant->"Italic"],
 " ",
 Cell[BoxData[
  FormBox[
   RowBox[{"\[Omega]", "(", "n", ")"}], TraditionalForm]], "InlineMath",
  ExpressionUUID->"74352fc1-3b25-4912-8ac7-4b78d2e89bb4"],
 " and ",
 Cell[BoxData[
  FormBox[
   RowBox[{"\[CapitalOmega]", "(", "n", ")"}], TraditionalForm]], 
  "InlineMath",ExpressionUUID->"40df1d49-009e-45ee-b7a8-7b0dc26ba8cb"],
 " count the number of prime factors of a natural number ",
 Cell[BoxData[
  FormBox["n", TraditionalForm]], "InlineMath",ExpressionUUID->
  "b1526a4e-0d32-4172-aea9-60a41395dd8d"],
 "."
}], "Text",
 CellID->1661812466,ExpressionUUID->"b8a8d00d-707d-42ac-bd7a-8ed41622ed3e"],

Cell[CellGroupData[{

Cell[TextData[{
 "Prime Omega Function ",
 Cell[BoxData[
  FormBox[
   RowBox[{"\[Omega]", "(", "n", ")"}], TraditionalForm]],
  FontSlant->"Plain",ExpressionUUID->"aa176e31-edf7-490c-985a-961dedac6823"]
}], "Subsubsection",
 CellID->1138430537,ExpressionUUID->"4ab62bf0-f6f1-4e76-9109-4914fa512335"],

Cell[TextData[{
 "The ",
 StyleBox["prime omega function",
  FontSlant->"Italic"],
 " ",
 Cell[BoxData[
  FormBox[
   RowBox[{"\[Omega]", "(", "n", ")"}], TraditionalForm]], "InlineMath",
  ExpressionUUID->"bb51bba6-c94c-4c38-ad39-65d8713d6e82"],
 " counts the number of distinct primes of ",
 Cell[BoxData[
  FormBox["n", TraditionalForm]], "InlineMath",ExpressionUUID->
  "791adfd2-1aca-4fe5-bcff-5543871fe833"],
 "."
}], "Text",
 CellID->2057286827,ExpressionUUID->"8890c490-5844-4b59-a520-b12cc302b565"],

Cell[TextData[{
 "If ",
 Cell[BoxData[
  FormBox[
   RowBox[{"n", "=", " ", 
    RowBox[{
     TemplateBox[{"p", "1", 
       SubscriptBox["k", "1"]},
      "Subsuperscript"], " ", 
     TemplateBox[{"p", "2", 
       SubscriptBox["k", "2"]},
      "Subsuperscript"], "\[Ellipsis]", " ", 
     TemplateBox[{"p", "m", 
       SubscriptBox["k", "m"]},
      "Subsuperscript"]}]}], TraditionalForm]], "InlineMath",ExpressionUUID->
  "797cbbf4-f30c-4f7b-a89a-a87d53ca165b"],
 ", then"
}], "Text",
 CellID->1503273659,ExpressionUUID->"d235a26f-63ea-467b-9cb6-e323cebf8e01"],

Cell[BoxData[
 FormBox[
  RowBox[{
   RowBox[{"\[Omega]", "(", "n", ")"}], "=", "m"}], 
  TraditionalForm]], "DisplayFormula",
 CellID->979255788,ExpressionUUID->"d839ea11-dd78-4497-9127-bb694722936d"],

Cell[TextData[{
 "Here is a short table of values of ",
 Cell[BoxData[
  FormBox[
   RowBox[{"\[Omega]", "(", "n", ")"}], TraditionalForm]], "InlineMath",
  ExpressionUUID->"32abe0d9-f127-40bb-96d3-fbd827bc96e6"],
 ":"
}], "Text",
 CellID->1717241097,ExpressionUUID->"4f4e8cf3-784c-4f86-aa98-ddfd72b5055f"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Grid", "[", 
  RowBox[{
   RowBox[{"Flatten", "/@", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"\"\<\!\(TraditionalForm\`n\)\>\"", ",", 
        RowBox[{"Range", "[", "10", "]"}]}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{"\"\<\!\(TraditionalForm\`\[Omega](n)\)\>\"", ",", 
        RowBox[{"Table", "[", 
         RowBox[{
          RowBox[{"PrimeNu", "[", "n", "]"}], ",", 
          RowBox[{"{", 
           RowBox[{"n", ",", "10"}], "}"}]}], "]"}]}], "}"}]}], "}"}]}], ",", 
   
   InterpretationBox[
    DynamicModuleBox[{Typeset`open = False}, 
     TemplateBox[{"Expression", "SequenceIcon", 
       GridBox[{{
          RowBox[{
            TagBox["\"Head: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            TagBox["Sequence", "IconizedItem"]}]}, {
          RowBox[{
            TagBox["\"Length: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            TagBox["2", "IconizedItem"]}]}, {
          RowBox[{
            TagBox["\"Byte count: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            
            TagBox["280", "IconizedItem"]}]}}, 
        GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> 
        "Column", 
        GridBoxItemSize -> {
         "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], 
       Dynamic[Typeset`open]},
      "IconizedObject"]],
    Sequence[Frame -> All, Background -> {None, {LightGray, LightBlue}}],
    SelectWithContents->True,
    Selectable->False]}], "]"}]], "Input",
 CellID->708577063,ExpressionUUID->"9c82d9f0-c9bf-482d-9fad-ffb5ff9aa5af"],

Cell[BoxData[
 TagBox[GridBox[{
    {"\<\"\\!\\(TraditionalForm\\`n\\)\"\>", "1", "2", "3", "4", "5", "6", 
     "7", "8", "9", "10"},
    {"\<\"\\!\\(TraditionalForm\\`\[Omega](n)\\)\"\>", "0", "1", "1", "1", 
     "1", "2", "1", "1", "1", "2"}
   },
   AutoDelete->False,
   GridBoxBackground->{"Columns" -> {{None}}, "Rows" -> {
       GrayLevel[0.85], 
       RGBColor[0.87, 0.94, 1]}},
   GridBoxFrame->{"Columns" -> {{True}}, "Rows" -> {{True}}},
   GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}],
  "Grid"]], "Output",
 CellChangeTimes->{3.936242083825328*^9},
 CellID->2121983046,ExpressionUUID->"342fa5cc-7e73-4377-9aa2-12c64437f74b"]
}, {2}]],

Cell[TextData[{
 "Here is the plot of ",
 Cell[BoxData[
  FormBox[
   RowBox[{"\[Omega]", "(", "n", ")"}], TraditionalForm]], "InlineMath",
  ExpressionUUID->"a6c6931c-9c6c-4e4c-ae0b-1b4aa7f3158e"],
 " for all numbers up to ",
 Cell[BoxData[
  FormBox["50", TraditionalForm]], "InlineMath",ExpressionUUID->
  "73cf9ee7-d8a1-fd4f-a522-343621fbf642"],
 ":"
}], "Text",
 CellID->827290817,ExpressionUUID->"2deef9ee-6d90-4265-8336-650d58d5561b"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"DiscretePlot", "[", 
  RowBox[{
   RowBox[{"PrimeNu", "[", "n", "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"n", ",", "50"}], "}"}], ",", 
   RowBox[{"PlotMarkers", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"Automatic", ",", "Small"}], "}"}]}], ",", 
   RowBox[{"AxesOrigin", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"0", ",", "0"}], "}"}]}], ",", 
   RowBox[{"ImageSize", "->", "Medium"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.936241753872827*^9, 3.9362417553485193`*^9}},
 CellID->786110471,ExpressionUUID->"6c9e938c-6214-40bc-ac1e-501f86ae9971"],

Cell[BoxData[
 GraphicsBox[{{
    {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.009166666666666668], 
     AbsoluteThickness[2], 
     TagBox[{
       {RGBColor[0.368417, 0.506779, 0.709798], PointSize[
        0.009166666666666668], AbsoluteThickness[2], Opacity[0.2], 
        LineBox[{{{1., 0.}, {1., 0.}}},
         VertexColors->None]}, 
       {RGBColor[0.368417, 0.506779, 0.709798], PointSize[
        0.009166666666666668], AbsoluteThickness[2], Opacity[0.2], 
        LineBox[CompressedData["
1:eJxd0zkKAkEQRuFCkwkNDAwmUBEREXHc15l219QjCMZeoY7sERTsUuo1DPLz
8aZFsHF/3h5lEck+Tyk+8YTvx6vw204CT+AVeAVehVfhNdsBO54UfQqvw+vw
JrwJb+H+FryNvg3voO/Au+i78B76HrwP78MHeP8AnsEz+BDvH8JH6EfwMfox
fAKfwKfwKXyG+2fwOfo5fIF+AV+iX8JX6FfwNfo1fIN+A8/jtv9VDi/QF/AA
D971twN2PFvv/x17uMJlh37nXW3H76dw2aPfe1e4wuWA/uBdbdv9cDnGbb//
0bvath4uJ9x/8q5whcsZ/dm72rb74XJBf/Gutq2HyxX9b8sb7wJZbQ==
         "],
         VertexColors->None]}},
      Annotation[#, "Charting`Private`Tag$79396#1"]& ]}, 
    TagBox[
     {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.009166666666666668],
       AbsoluteThickness[2], {}, GeometricTransformationBox[InsetBox[
        GraphicsBox[
         {EdgeForm[None], DiskBox[{0, 0}]},
         DefaultBaseStyle->{"Graphics", 
           Directive[
            PointSize[0.009166666666666668], 
            RGBColor[0.368417, 0.506779, 0.709798], 
            AbsoluteThickness[2]]},
         PlotRangePadding->Scaled[0.15]], {0, 0}, Automatic, Offset[
        7]], CompressedData["
1:eJxd0bsOAUEUh/GDRqlQKBSIiIgI637dcVlW6xEkaq8wj+wRbOEvOd8kU3z7
mzOZZNvP9+NVMbOk2KVil03rk5pfwX+vomvoOroR/D1NeAvdQXcx34P34QP4
ED5Cj3F+gk5wfgqfwefoBXqJ+RV8Dd/At/AdfA8//Fr/LYUH31Gt+46+I9pO
cLXuP8PRdoGrNZ/598cMfsU82m5wteZzeA6///0LuNY/uw==
       "]], {}},
     Annotation[#, "Charting`Private`Tag$79396#1"]& ]}, {}, {}},
  AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
  Axes->True,
  AxesOrigin->{0., 0.},
  DisplayFunction->Identity,
  FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
  GridLines->{None, None},
  GridLinesStyle->Directive[
    GrayLevel[0.5, 0.4]],
  ImageSize->Medium,
  Method->{
   "MessagesHead" -> DiscretePlot, "AxisPadding" -> Scaled[0.02], 
    "DefaultBoundaryStyle" -> Automatic, 
    "DefaultGraphicsInteraction" -> {
     "Version" -> 1.2, "TrackMousePosition" -> {True, False}, 
      "Effects" -> {
       "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, 
        "Droplines" -> {
         "freeformCursorMode" -> True, 
          "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" -> 
    AbsolutePointSize[6], "DefaultPlotStyle" -> {
      Directive[
       RGBColor[0.368417, 0.506779, 0.709798], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.880722, 0.611041, 0.142051], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.560181, 0.691569, 0.194885], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.922526, 0.385626, 0.209179], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.528488, 0.470624, 0.701351], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.772079, 0.431554, 0.102387], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.363898, 0.618501, 0.782349], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[1, 0.75, 0], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.647624, 0.37816, 0.614037], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.571589, 0.586483, 0.], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.915, 0.3325, 0.2125], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.40082222609352647`, 0.5220066643438841, 0.85], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.9728288904374106, 0.621644452187053, 0.07336199581899142], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.736782672705901, 0.358, 0.5030266573755369], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.28026441037696703`, 0.715, 0.4292089322474965], 
       AbsoluteThickness[2]]}, "DomainPadding" -> Scaled[0.02], 
    "PointSizeFunction" -> "SmallPointSize", "RangePadding" -> Scaled[0.05]},
  PlotRange->{{0, 50}, {0, 3.}},
  PlotRangeClipping->True,
  PlotRangePadding->{{
     Scaled[0.02], 
     Scaled[0.02]}, {
     Scaled[0.05], 
     Scaled[0.05]}},
  Ticks->{Automatic, Automatic}]], "Output",
 CellChangeTimes->{3.936241755798176*^9},
 CellID->2008861789,ExpressionUUID->"bad2157e-7657-42a3-b246-90f20e91c803"]
}, {2}]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
 "Prime Omega Function ",
 Cell[BoxData[
  FormBox[
   RowBox[{"\[CapitalOmega]", 
    RowBox[{"(", "n", ")"}]}], TraditionalForm]],
  FontSlant->"Plain",ExpressionUUID->"f01d3bf7-daa3-9b42-a456-2776dfaeeee6"]
}], "Subsubsection",
 CellChangeTimes->{{3.7929945775465393`*^9, 3.7929945894597263`*^9}, {
  3.7933345298281517`*^9, 3.7933345442659006`*^9}, {3.79524658291366*^9, 
  3.7952465854312215`*^9}, {3.796438341389743*^9, 3.7964383436971626`*^9}, {
  3.7964386574323964`*^9, 3.796438665638298*^9}, {3.889337354657016*^9, 
  3.8893373550935893`*^9}},
 CellID->764736412,ExpressionUUID->"72b5ef1f-2a7d-4104-b43c-c6370dca4210"],

Cell[TextData[{
 "The ",
 StyleBox["prime omega function ",
  FontSlant->"Italic"],
 Cell[BoxData[
  FormBox[
   RowBox[{"\[CapitalOmega]", "(", "n", ")"}], TraditionalForm]], 
  "InlineMath",ExpressionUUID->"e2a13c7c-cd11-4dac-bd8f-3795b3307157"],
 " counts the total number of prime factors of ",
 Cell[BoxData[
  FormBox["n", TraditionalForm]], "InlineMath",ExpressionUUID->
  "258ffcd1-4121-4407-8984-89122db479ec"],
 ", considering their multiplicity:"
}], "Text",
 CellID->235249189,ExpressionUUID->"9b27b8f0-7619-4540-9501-e31508c04b75"],

Cell[TextData[{
 "If ",
 Cell[BoxData[
  FormBox[
   RowBox[{"n", "=", " ", 
    RowBox[{
     TemplateBox[{"p", "1", 
       SubscriptBox["k", "1"]},
      "Subsuperscript"], " ", 
     TemplateBox[{"p", "2", 
       SubscriptBox["k", "2"]},
      "Subsuperscript"], "\[Ellipsis]", " ", 
     TemplateBox[{"p", "m", 
       SubscriptBox["k", "m"]},
      "Subsuperscript"]}]}], TraditionalForm]], "InlineMath",ExpressionUUID->
  "c649b09e-71ba-4e26-aed5-caa411cde9b9"],
 ", then"
}], "Text",
 CellID->898568374,ExpressionUUID->"36c778f2-2ab8-4729-b524-5d749f4f3957"],

Cell[BoxData[
 FormBox[
  RowBox[{
   RowBox[{"\[CapitalOmega]", "(", "n", ")"}], "=", 
   RowBox[{
    SubscriptBox["k", "1"], "+", "\[Ellipsis]", "+", 
    SubscriptBox["k", "m"]}]}], TraditionalForm]], "DisplayFormula",
 CellID->883391814,ExpressionUUID->"024f0371-9810-40ac-ab49-fd718e90db3a"],

Cell[TextData[{
 "Here is a short table of values of ",
 Cell[BoxData[
  FormBox[
   RowBox[{"\[CapitalOmega]", "(", "n", ")"}], TraditionalForm]], 
  "InlineMath",ExpressionUUID->"5f60e15e-9f8d-47fc-b934-61f302a24265"],
 ":"
}], "Text",
 CellID->537870064,ExpressionUUID->"2fdaaf86-7c01-428e-a36f-d31f95b252ab"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Grid", "[", 
  RowBox[{
   RowBox[{"Flatten", "/@", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"\"\<\!\(TraditionalForm\`n\)\>\"", ",", 
        RowBox[{"Range", "[", "10", "]"}]}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{"\"\<\!\(TraditionalForm\`\(TraditionalForm\`\[CapitalOmega](n)\
\)\)\>\"", ",", 
        RowBox[{"Table", "[", 
         RowBox[{
          RowBox[{"PrimeOmega", "[", "n", "]"}], ",", 
          RowBox[{"{", 
           RowBox[{"n", ",", "10"}], "}"}]}], "]"}]}], "}"}]}], "}"}]}], ",", 
   
   InterpretationBox[
    DynamicModuleBox[{Typeset`open = False}, 
     TemplateBox[{"Expression", "SequenceIcon", 
       GridBox[{{
          RowBox[{
            TagBox["\"Head: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            TagBox["Sequence", "IconizedItem"]}]}, {
          RowBox[{
            TagBox["\"Length: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            TagBox["2", "IconizedItem"]}]}, {
          RowBox[{
            TagBox["\"Byte count: \"", "IconizedLabel"], "\[InvisibleSpace]", 
            
            TagBox["280", "IconizedItem"]}]}}, 
        GridBoxAlignment -> {"Columns" -> {{Left}}}, DefaultBaseStyle -> 
        "Column", 
        GridBoxItemSize -> {
         "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], 
       Dynamic[Typeset`open]},
      "IconizedObject"]],
    Sequence[Frame -> All, Background -> {None, {LightGray, LightBlue}}],
    SelectWithContents->True,
    Selectable->False]}], "]"}]], "Input",
 CellChangeTimes->{3.9362421102757387`*^9},
 CellID->1223495425,ExpressionUUID->"6adaa962-9701-4172-aceb-a701abd2656f"],

Cell[BoxData[
 TagBox[GridBox[{
    {"\<\"\\!\\(TraditionalForm\\`n\\)\"\>", "1", "2", "3", "4", "5", "6", 
     "7", "8", "9", "10"},
    {"\<\"\\!\\(TraditionalForm\\`\\(TraditionalForm\\`\[CapitalOmega](n)\\)\\\
)\"\>", "0", "1", "1", "2", "1", "2", "1", "3", "2", "2"}
   },
   AutoDelete->False,
   GridBoxBackground->{"Columns" -> {{None}}, "Rows" -> {
       GrayLevel[0.85], 
       RGBColor[0.87, 0.94, 1]}},
   GridBoxFrame->{"Columns" -> {{True}}, "Rows" -> {{True}}},
   GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}],
  "Grid"]], "Output",
 CellChangeTimes->{
  3.797582138587577*^9, 3.935740153494761*^9, {3.936242111326048*^9, 
   3.9362421169042273`*^9}},
 CellID->865604348,ExpressionUUID->"546a82b1-232e-4b31-84dc-64b75f62d5ee"]
}, {2}]],

Cell[TextData[{
 "Here is the plot of ",
 Cell[BoxData[
  FormBox[
   RowBox[{"\[CapitalOmega]", "(", "n", ")"}], TraditionalForm]],
  ExpressionUUID->"f2c7c90a-6b7c-4394-a0a2-5bce2de0720b"],
 " for all numbers up to ",
 Cell[BoxData[
  FormBox["50", TraditionalForm]], "InlineMath",ExpressionUUID->
  "12f79b84-5a46-48f4-ae5f-52ead41d56fd"],
 ":"
}], "Text",
 CellID->16714171,ExpressionUUID->"61412fa3-8a2e-4f5b-9383-2d145bad36e6"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"DiscretePlot", "[", 
  RowBox[{
   RowBox[{"PrimeOmega", "[", "n", "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"n", ",", "50"}], "}"}], ",", 
   RowBox[{"PlotMarkers", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"Automatic", ",", "Small"}], "}"}]}], ",", 
   RowBox[{"AxesOrigin", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"0", ",", "0"}], "}"}]}], ",", 
   RowBox[{"ImageSize", "->", "Medium"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.9362417735242653`*^9, 3.9362417753129807`*^9}},
 CellID->1435373755,ExpressionUUID->"0448da14-1336-4b1e-8b84-4420a80d21bd"],

Cell[BoxData[
 GraphicsBox[{{
    {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.009166666666666668], 
     AbsoluteThickness[2], 
     TagBox[{
       {RGBColor[0.368417, 0.506779, 0.709798], PointSize[
        0.009166666666666668], AbsoluteThickness[2], Opacity[0.2], 
        LineBox[{{{1., 0.}, {1., 0.}}},
         VertexColors->None]}, 
       {RGBColor[0.368417, 0.506779, 0.709798], PointSize[
        0.009166666666666668], AbsoluteThickness[2], Opacity[0.2], 
        LineBox[CompressedData["
1:eJxd07tqQlEQRuEhaSxTWFhYGBEREfEW73q215g2jxBInVeYR84jJJD5DbM2
iCw+xtke8fnj6/3z0czGv6+HeMUpf2/fVW6dGrwGf1IXdJw65uvwBuYb8Cbm
m/BWyfdqwdv4/Da8A+/Au9jfhfewvwfvY74PH2D/AD6ED+GjaD33EXyM/WP4
JFr3n8CnmJ/CZ5ifwV/UBR1nDp/DF9i/gC+j9f2X8BU+fwVfw9fwTbS+3wa+
hW/hu2jdfwffY34PrzBfwUu0/lclu9+7oOMcsv93zMMdbsdoPf9jdlfH/R1u
J+w/ZXe4w+2M/efsrtZ+uF2i9fwv2V2tebhdMX/N7nCH26u6oGNerf1wu0Xr
979l93sXdJy3fL97m/0APDNSpQ==
         "],
         VertexColors->None]}},
      Annotation[#, "Charting`Private`Tag$79421#1"]& ]}, 
    TagBox[
     {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.009166666666666668],
       AbsoluteThickness[2], {}, GeometricTransformationBox[InsetBox[
        GraphicsBox[
         {EdgeForm[None], DiskBox[{0, 0}]},
         DefaultBaseStyle->{"Graphics", 
           Directive[
            PointSize[0.009166666666666668], 
            RGBColor[0.368417, 0.506779, 0.709798], 
            AbsoluteThickness[2]]},
         PlotRangePadding->Scaled[0.15]], {0, 0}, Automatic, Offset[
        7]], CompressedData["
1:eJxdkbsKAkEMRaM2lhYWFhYqIiIiur4f68bH6tr6CYK1vzCf7Ce4xUTICUxx
5ubeTJju6/N810QkKU+lPFWx+mbiS/19HdxQ39eE3oLeht5Rn9tDfx88gH8I
/wj6GP4JeKp+jxn8CfLn0BfQl8hfgdfwbzB/i/4deI95B3CK/CP0DLpGjv8W
jG3eyXMAy9m/Pxhb/gV+sFzhNzZ/7t8fcug36GC5Y76x+YvItn/h++Xxz/sB
Duo48w==
       "]], {}},
     Annotation[#, "Charting`Private`Tag$79421#1"]& ]}, {}, {}},
  AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
  Axes->True,
  AxesOrigin->{0., 0.},
  DisplayFunction->Identity,
  FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
  GridLines->{None, None},
  GridLinesStyle->Directive[
    GrayLevel[0.5, 0.4]],
  ImageSize->Medium,
  Method->{
   "MessagesHead" -> DiscretePlot, "AxisPadding" -> Scaled[0.02], 
    "DefaultBoundaryStyle" -> Automatic, 
    "DefaultGraphicsInteraction" -> {
     "Version" -> 1.2, "TrackMousePosition" -> {True, False}, 
      "Effects" -> {
       "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, 
        "Droplines" -> {
         "freeformCursorMode" -> True, 
          "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" -> 
    AbsolutePointSize[6], "DefaultPlotStyle" -> {
      Directive[
       RGBColor[0.368417, 0.506779, 0.709798], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.880722, 0.611041, 0.142051], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.560181, 0.691569, 0.194885], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.922526, 0.385626, 0.209179], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.528488, 0.470624, 0.701351], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.772079, 0.431554, 0.102387], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.363898, 0.618501, 0.782349], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[1, 0.75, 0], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.647624, 0.37816, 0.614037], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.571589, 0.586483, 0.], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.915, 0.3325, 0.2125], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.40082222609352647`, 0.5220066643438841, 0.85], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.9728288904374106, 0.621644452187053, 0.07336199581899142], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.736782672705901, 0.358, 0.5030266573755369], 
       AbsoluteThickness[2]], 
      Directive[
       RGBColor[0.28026441037696703`, 0.715, 0.4292089322474965], 
       AbsoluteThickness[2]]}, "DomainPadding" -> Scaled[0.02], 
    "PointSizeFunction" -> "SmallPointSize", "RangePadding" -> Scaled[0.05]},
  PlotRange->{{0, 50}, {0, 5.}},
  PlotRangeClipping->True,
  PlotRangePadding->{{
     Scaled[0.02], 
     Scaled[0.02]}, {
     Scaled[0.05], 
     Scaled[0.05]}},
  Ticks->{Automatic, Automatic}]], "Output",
 CellChangeTimes->{3.936241775759026*^9},
 CellID->226090980,ExpressionUUID->"e138b246-b3f0-41b0-b5d1-86f6ab56968b"]
}, {2}]],

Cell[CellGroupData[{

Cell["Summary", "TechNoteSection",
 CellID->1259673378,ExpressionUUID->"436f7a98-fb5b-4a7c-8adc-1755eff36ad0"],

Cell["Every integer is a product of primes.", "SummaryList",
 CellChangeTimes->{
  3.7964490725818167`*^9, {3.796449126718799*^9, 3.7964491284698105`*^9}},
 CellID->606253025,ExpressionUUID->"a7178d9f-5359-4289-b355-f74602787985"],

Cell["\<\
The primes in the product are neither distinct nor arranged in any specific \
order.\
\>", "SummaryList",
 CellChangeTimes->{{3.7961182351125965`*^9, 3.796118238797512*^9}, {
  3.7961186029445457`*^9, 3.796118607792718*^9}, {3.7961186549317284`*^9, 
  3.796118686221808*^9}, {3.79620001806293*^9, 3.796200046494235*^9}, {
  3.7962002150997863`*^9, 3.7962002164727244`*^9}, {3.796200263723717*^9, 
  3.796200277575384*^9}, {3.8789610084528713`*^9, 3.8789610541603913`*^9}, {
  3.8789610990740013`*^9, 3.878961108985112*^9}, {3.8789617180658197`*^9, 
  3.8789617213019085`*^9}, {3.87899102762439*^9, 3.878991038187935*^9}, {
  3.8839991630646744`*^9, 3.883999167026672*^9}},
 CellID->1632169807,ExpressionUUID->"987b8e00-84e3-4d07-a3d7-9f0ecaa11b1f"],

Cell[TextData[{
 "The ",
 StyleBox["fundamental theorem of arithmetic",
  FontSlant->"Italic"],
 StyleBox[" ",
  FontWeight->"Bold"],
 StyleBox["states",
  FontWeight->"Plain"],
 " that every integer greater than ",
 Cell[BoxData[
  FormBox["1", TraditionalForm]], "InlineMath",ExpressionUUID->
  "9b8b37ac-8c43-4115-a93a-f2380970b1b6"],
 " can be represented as a product of prime factors in only one way, apart \
from the order of the factors."
}], "SummaryList",
 CellID->2069323118,ExpressionUUID->"65134c60-012f-4597-952e-f51dd41378b2"],

Cell[TextData[{
 "There are some very important functions connected with ",
 StyleBox["prime factorization",
  FontSlant->"Italic"],
 "."
}], "SummaryList",
 CellChangeTimes->{{3.796449147328994*^9, 3.7964491680621166`*^9}, {
  3.8878825552476664`*^9, 3.887882555509231*^9}, {3.9430237139270573`*^9, 
  3.943023714409376*^9}},
 CellID->615981061,ExpressionUUID->"eb9e406d-b2a4-441d-96be-711fe1d88ca6"]
}, Open  ]]
}, Open  ]]
}, Open  ]]
}, Open  ]],

Cell["\[Copyright] 2026 Wolfram Media, Inc.", "SmallText",
 CellMargins->{{66, 66}, {20, 100}},
 CellGroupingRules->"SectionGrouping",
 FontSize->12,
 FontColor->RGBColor[
  0.7098039215686275, 0.7098039215686275, 
   0.7098039215686275],ExpressionUUID->"335b811d-f0d1-4160-8ae7-5c3a9270b7f0"]
},
Editable->True,
Saveable->False,
ScreenStyleEnvironment->"Working",
WindowToolbars->{},
CellInsertionPointCell->None,
WindowSize->Automatic,
WindowMargins->Automatic,
WindowFrame->"Normal",
WindowPersistentStyles->{
 WindowFrame, WindowMargins, WindowSize, Magnification},
WindowTitle->"4. Prime Factorization | Introduction to Number Theory",
DockedCells->{
  Cell[
   BoxData[
    TemplateBox[<|
     "label" -> TemplateBox[<|
        "heading" -> "Wolfram eTextbook Series", "title" -> 
         "Introduction to Number Theory"|>, "BannerHeadingTitleTemplate"], 
      "previous" -> TemplateBox[{
         TemplateBox[{
           TemplateBox[{}, "BannerLeftArrowTemplate"], 
           Function[{}, 
            
            NotebookLocate[{"NumberTheory-01b_Exercises.nb", ""}, 
             FrontEnd`HistoryData :> {
               EvaluationNotebook[]}]]}, "BannerButtonTemplate"], 
         "\nPart 1 Exercises"}, "PrettyTooltipTemplate"], "next" -> 
      TemplateBox[{
         TemplateBox[{
           TemplateBox[{}, "BannerRightArrowTemplate"], 
           Function[{}, 
            
            NotebookLocate[{
             "NumberTheory-02-05_Multiples-and-Divisors.nb", ""}, 
             FrontEnd`HistoryData :> {
               EvaluationNotebook[]}]]}, "BannerButtonTemplate"], 
         "\n5. Multiples and Divisors"}, "PrettyTooltipTemplate"], "menu" -> 
      TemplateBox[{
         TemplateBox[{{
           "Cover" :> NotebookLocate[{"NumberTheory-00a_Cover.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "Preface" :> 
            NotebookLocate[{"NumberTheory-00b_Preface.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], StyleBox["Introduction", Bold] :> 
            NotebookLocate[{
              "NumberTheory-01-01_What-Is-Number-Theory.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "1. What Is Number Theory?" :> 
            NotebookLocate[{
              "NumberTheory-01-01_What-Is-Number-Theory.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "2. Integers: The Basics" :> 
            NotebookLocate[{"NumberTheory-01-02_Integers-The-Basics.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "3. Primes and Composites" :> 
            NotebookLocate[{
              "NumberTheory-01-03_Primes-and-Composites.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "Part 1 Exercises" :> 
            NotebookLocate[{"NumberTheory-01b_Exercises.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], 
            StyleBox["Primes and Divisibility", Bold] :> 
            NotebookLocate[{"NumberTheory-02-04_Prime-Factorization.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "4. Prime Factorization" :> 
            NotebookLocate[{"NumberTheory-02-04_Prime-Factorization.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "5. Multiples and Divisors" :> 
            NotebookLocate[{
              "NumberTheory-02-05_Multiples-and-Divisors.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "6. Greatest Common Divisor" :> 
            NotebookLocate[{
              "NumberTheory-02-06_Greatest-Common-Divisor.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "Part 2 Exercises" :> 
            NotebookLocate[{"NumberTheory-02b_Exercises.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], StyleBox["Modular Arithmetic", Bold] :> 
            NotebookLocate[{"NumberTheory-03-07_Congruences.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "7. Congruences" :> 
            NotebookLocate[{"NumberTheory-03-07_Congruences.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "8. Modular Arithmetic" :> 
            NotebookLocate[{"NumberTheory-03-08_Modular-Arithmetic.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "9. Chinese Remainder Theorem" :> 
            NotebookLocate[{
              "NumberTheory-03-09_Chinese-Remainder-Theorem.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "Part 3 Exercises" :> 
            NotebookLocate[{"NumberTheory-03b_Exercises.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], StyleBox["Continued Fractions", Bold] :> 
            NotebookLocate[{
              "NumberTheory-04-10_Real-Number-Representations.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "10. Real Number Representations" :> 
            NotebookLocate[{
              "NumberTheory-04-10_Real-Number-Representations.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "11. Continued Fractions" :> 
            NotebookLocate[{"NumberTheory-04-11_Continued-Fractions.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "12. Best Rational Approximations" :> 
            NotebookLocate[{
              "NumberTheory-04-12_Best-Rational-Approximations.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "Part 4 Exercises" :> 
            NotebookLocate[{"NumberTheory-04b_Exercises.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], 
            StyleBox["Diophantine Equations", Bold] :> 
            NotebookLocate[{
              "NumberTheory-05-13_Diophantine-Equations.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "13. Diophantine Equations" :> 
            NotebookLocate[{
              "NumberTheory-05-13_Diophantine-Equations.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "14. Linear Diophantine Equations" :> 
            NotebookLocate[{
              "NumberTheory-05-14_Linear-Diophantine-Equations.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], 
            "15. Diophantine Equations of Degree 2" :> 
            NotebookLocate[{
              "NumberTheory-05-15_Diophantine-Equations-of-Degree-2.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "Part 5 Exercises" :> 
            NotebookLocate[{"NumberTheory-05b_Exercises.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], StyleBox["Cryptography", Bold] :> 
            NotebookLocate[{"NumberTheory-06-16_Cryptography.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "16. Cryptography" :> 
            NotebookLocate[{"NumberTheory-06-16_Cryptography.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "17. The RSA Algorithm" :> 
            NotebookLocate[{"NumberTheory-06-17_The-RSA-Algorithm.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "Part 6 Exercises" :> 
            NotebookLocate[{"NumberTheory-06b_Exercises.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], StyleBox["Additional Topics", Bold] :> 
            NotebookLocate[{
              "NumberTheory-07-18_Primality-Testing-Miller-Rabin.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], 
            "18. Primality Testing: Miller\[Dash]Rabin" :> 
            NotebookLocate[{
              "NumberTheory-07-18_Primality-Testing-Miller-Rabin.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], 
            "19. Factorization: Pollard\[CloseCurlyQuote]s Rho" :> 
            NotebookLocate[{
              "NumberTheory-07-19_Factorization-Pollards-Rho.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], 
            "20. Diophantine Equations: Hilbert\[CloseCurlyQuote]s Tenth \
Problem" :> NotebookLocate[{
              "NumberTheory-07-20_Diophantine-Equations-Hilberts-Tenth-\
Problem.nb", ""}, FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "21. The Riemann Zeta Function" :> 
            NotebookLocate[{
              "NumberTheory-07-21_The-Riemann-Zeta-Function.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "22. Special Numbers" :> 
            NotebookLocate[{"NumberTheory-07-22_Special-Numbers.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "23. Numeral Systems" :> 
            NotebookLocate[{"NumberTheory-07-23_Numeral-Systems.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}], "Part 7 Exercises" :> 
            NotebookLocate[{"NumberTheory-07b_Exercises.nb", ""}, 
              FrontEnd`HistoryData :> {
                EvaluationNotebook[]}]}}, "BannerMenuTemplate"], 
         "Table of contents"}, "PrettyTooltipTemplate"]|>, "BannerTemplate"]],
    "DockedCell", "Banner"], 
  Cell[
   BoxData[
    TemplateBox[<|
     "label" -> "Lusine Sukiasyan and Charles Pooh  |  Cite this eTextbook"|>,
      "BannerCitationOpenerTemplate"]], "DockedCell", 
   "BannerCitationOpenerLabel"], 
  Cell[
   BoxData[
    TemplateBox[{
      TemplateBox[{"\"Lusine Sukiasyan and Charles Pooh (2026), \"", 
        StyleBox[
        "\"Introduction to Number Theory: A Computational Approach,\"", 
         Selectable -> True, Copyable -> True, FontSlant -> Italic], 
        "\" Wolfram Media. \
wolfram-media.com/products/introduction-to-number-theory.\""}, "RowDefault"], 
      
      RowBox[{"Lusine Sukiasyan and Charles Pooh (2026), ", 
        StyleBox[
        "Introduction to Number Theory: A Computational Approach,", FontSlant -> 
         "Italic"], 
        " Wolfram Media. \
wolfram-media.com/products/introduction-to-number-theory."}]}, 
     "CitationClickToCopy2", BaseStyle -> {LineBreakWithin -> Automatic}]], 
   "DockedCell", "BannerCitation", CellOpen -> Dynamic[
     TrueQ[
      CurrentValue[
       EvaluationNotebook[], {TaggingRules, "eTextbook", "CitationOpener"}, 
       False]]]]},
PrintingCopies->1,
PrintingPageRange->{32000, 32000},
PrintingOptions->{"Magnification"->1.,
"PaperOrientation"->"Portrait",
"PaperSize"->{612, 792}},
ShowCellBracket->Automatic,
TaggingRules-><|"eTextbook" -> <|"CitationOpener" -> False|>|>,
ShowCellLabel->False,
CellLabelAutoDelete->True,
CellLabelTemplate-><|
 "In" -> "In[`1`]:=", "InExpired" -> "In[`1`]:=", "Out" -> "Out[`1`]=", 
  "OutExpired" -> "Out[`1`]="|>,
ShowCellTags->False,
ShowAutoSpellCheck->False,
ShowPredictiveInterface->False,
AutoStyleOptions->{"HighlightComments"->True},
AutoMultiplicationSymbol->{},
Magnification->1,
FrontEndVersion->"14.3 for Mac OS X ARM (64-bit) (July 1, 2025)",
StyleDefinitions->Notebook[{
   Cell[
    StyleData[StyleDefinitions -> "Default.nb"]], 
   Cell[
    CellGroupData[{
      Cell[
      "Style Environments", "Section", 
       CellChangeTimes -> {{3.64661165091824*^9, 3.646611654013329*^9}, 
         3.6485723794335403`*^9, 3.651347081728628*^9, 3.651347179462186*^9, 
         3.651347384135805*^9, 3.651348171662589*^9}], 
      Cell[
       StyleData[All, "Working"], CellInsertionPointCell -> 
       FEPrivate`FrontEndResource["FEExpressions", "CellInsertionMenu"], 
       DockedCells -> {
         Cell[
          BoxData[
           GridBox[{{
              InterpretationBox[
               StyleBox[
                
                GraphicsBox[{}, BaselinePosition -> Baseline, 
                 ImageSize -> {4, 0}], CacheGraphics -> False], 
               Spacer[5]], 
              ActionMenuBox[
              "\"Insert\"", {
               "\"Subsection\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTConvertToSubsectionStyle[]), 
                "\"Subsubsection\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTConvertToSubsubsectionStyle[]), 
                "\"Subsubsubsection\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTConvertToSubsubsubsectionStyle[]), 
                Delimiter, 
                "\"Text\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTConvertToTextStyle[]), "\"CodeText\"" :> 
                Module[{$CellContext`bn = ButtonNotebook[]}, 
                  If[Developer`CellInformation[$CellContext`bn] =!= $Failed, 
                    While[
                    FreeQ[
                    Developer`CellInformation[$CellContext`bn], 
                    "CellBracket"], 
                    FrontEndExecute[
                    
                    FrontEnd`SelectionMove[$CellContext`bn, All, Cell, 
                    AutoScroll -> False]]]; 
                    SelectionMove[$CellContext`bn, After, Cell]]; 
                  NotebookWrite[$CellContext`bn, 
                    Cell["XXXX", "CodeText"], All]; 
                  SelectionMove[$CellContext`bn, All, CellContents]], 
                "\"Hyperlink\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTGenerateHyperlinkDialog[]), 
                "\"Display Formula\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertDisplayFormulaCell[]), Delimiter, 
                "\"Solved Example\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`\
WBTInsertSolvedExampleAndSolvedExampleNoteCells[]), 
                "\"Solved Example Input\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertSolvedExampleInputCell[]), 
                "\"Solved Example Text\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertSolvedExampleNoteCell[]), 
                "\"Solved Example Display Formula\"" :> (
                 Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertSolvedExampleDisplayFormulaCell[]),
                 "\"Solved Example Display Formula Numbered\"" :> (
                 Needs["WolframBookTools`"]; 
                 WolframBookTools`\
WBTInsertSolvedExampleDisplayFormulaNumberedCell[]), Delimiter, 
                "\"Theorem\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertTheoremCell[]), 
                "\"Theorem Numbered\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertTheoremNumberedCell[]), 
                "\"Theorem Text\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertTheoremTextCell[]), 
                "\"Proof/Theorem Display Formula\"" :> (
                 Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertProofTheoremDisplayCell[]), 
                "\"Proof/Theorem Display Formula Numbered\"" :> (
                 Needs["WolframBookTools`"]; 
                 WolframBookTools`\
WBTInsertProofTheoremDisplayFormulaNumberedCell[]), 
                "\"Proof\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertProofCell[]), 
                "\"Proof Text\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertProofTextCell[]), Delimiter, 
                "\"Vocabulary 2Col\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertVocabularySectionAnd2ColTable[]), 
                "\"Vocabulary 3Col\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertVocabularySectionAnd3ColTable[]), 
                "\"Add Row\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTVocabTableAddRow[]), 
                "\"Remove Row\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTVocabTableAddRow["RemoveRow" -> True]), 
                "\"Apply Divider\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTVocabTableAddRow["AddDivider" -> True]), 
                "\"Remove Divider\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTVocabTableAddRow[
                  "RemoveDivider" -> True]), Delimiter, 
                "\"Exercises\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertExerciseSection[]), 
                "\"Exercise\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertExerciseSection[]), 
                "\"Exercise Note\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertExerciseNote[]), 
                "\"Solution Section\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertExerciseSolution[]), 
                "\"Solution Text\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertSolutionAnswer[]), Delimiter, 
                "\"Q&A | Tech Notes | More\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertQAndATechNotesMore[]), 
                "\"Question\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertQuestion[]), 
                "\"Answer\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertAnswer[]), 
                "\"Tech Note\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertTechNote[]), 
                "\"More to Explore\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertMoreExplore[]), Delimiter, 
                "\"References Section\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertReferencesSection[]), 
                "\"Add Reference\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTAddReferenceCell[]), Delimiter, 
                "\"Comment\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTApplyComment[])}, FrameMargins -> 2.4], 
              ItemBox["", ItemSize -> Fit], 
              ActionMenuBox[
              "\"Function Name\"", {
               "\"Link\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTMakeSymbolLink[]), 
                "\"Format\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTFormatSymbol[])}, Appearance -> 
               "PopupMenu", Method -> "Queued", FrameMargins -> 2.4], 
              ButtonBox[
              "\"Resource Function Link\"", 
               ButtonFunction :> (Needs["WolframBookTools`"]; 
                WolframBookTools`WBTResourceFunctionLink[]), Evaluator -> 
               Automatic, Appearance -> "DialogBox", Method -> "Queued"], 
              ActionMenuBox[
              "\"Inline Format\"", {
               "\"Inline Code\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTConvertToInlineCode[]), 
                "\"Traditional Math\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTTraditionalFormCell[])}, Appearance -> 
               "PopupMenu", Method -> "Queued", FrameMargins -> 2.4], 
              ActionMenuBox[
              "\"Mark Cells\"", {
               "\"Tentative\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTApplyTentative[]), Delimiter, 
                "\"Print Only\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTApplyPrintOnly[]), 
                "\"Web Only\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTApplyWebOnly[]), Delimiter, 
                "\"Locked\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTApplyLock[])}, Appearance -> "PopupMenu",
                Method -> "Queued", FrameMargins -> 2.4], 
              ItemBox["", ItemSize -> Fit], 
              GridBox[{{
                 TooltipBox[
                  ButtonBox[
                  "\"Index Field\"", Appearance -> {"AbuttingRight"}, 
                   ButtonFunction :> (Needs["WolframBookTools`"]; 
                    WolframBookTools`WBTInsertIndexField[]), Evaluator -> 
                   Automatic, Method -> "Queued"], 
                  "First put the cursor in a cell to insert a corresponding \
field above to insert index entries.\nList main index entries corresponding \
to the cell separated by commas. Any associated sub-entries\nshould be put \
immediately after in braces.\nExample: programming{functional, rule based}, \
function, relation, lists{flat, nested}\n If $Language is \"ChineseSimplified\
\", \"ChineseTraditional\", \"Japanese\", \"Korean\" or \"Russian\"\ntwo \
index fields get inserted above a selected cell, the top one for inserting \
entries\nintended for the Englsh index and the bottom for the index in the \
language given by $Language.", TooltipDelay -> 0.3], 
                 TooltipBox[
                  ButtonBox[
                  "\"Toggle\"", Appearance -> {"AbuttingLeft"}, 
                   ButtonFunction :> (CurrentValue[
                    ButtonNotebook[], ScreenStyleEnvironment] = 
                    If[CurrentValue[
                    ButtonNotebook[], ScreenStyleEnvironment] === 
                    "Invisibles", "Working", "Invisibles"]), Evaluator -> 
                   Automatic, Method -> "Queued"], 
                  "Toggles index entry fields.", TooltipDelay -> 0.3]}}, 
               GridBoxItemSize -> {
                "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, 
               GridBoxSpacings -> {
                "ColumnsIndexed" -> {2 -> 0}, "Rows" -> {{Automatic}}}], 
              InterpretationBox[
               StyleBox[
                
                GraphicsBox[{}, BaselinePosition -> Baseline, 
                 ImageSize -> {4, 0}], CacheGraphics -> False], 
               Spacer[4]]}}]], "DockedCell", 
          CellFrameMargins -> {{0, 0}, {8, 8}}]}, CreateCellID -> True, 
       TrackCellChangeTimes -> False, ShowPredictiveInterface -> False, 
       AutoStyleOptions -> {
        "UndefinedSymbolStyle" -> 
         None}, $CellContext`GhostCellInEmptyNotebook -> False], 
      Cell[
       StyleData[All, "Printout"], PageWidth :> 700, 
       PrintingOptions -> {"PageSize" -> {504, 720}}, 
       AutoStyleOptions -> {"UndefinedSymbolStyle" -> None}, Magnification -> 
       0.6], 
      Cell[
       StyleData[All, "Invisibles"], CellInsertionPointCell -> 
       FEPrivate`FrontEndResource["FEExpressions", "CellInsertionMenu"], 
       DockedCells -> {
         Cell[
          BoxData[
           GridBox[{{
              InterpretationBox[
               StyleBox[
                
                GraphicsBox[{}, BaselinePosition -> Baseline, 
                 ImageSize -> {4, 0}], CacheGraphics -> False], 
               Spacer[4]], 
              ActionMenuBox[
              "\"Insert\"", {
               "\"Subsection\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTConvertToSubsectionStyle[]), 
                "\"Subsubsection\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTConvertToSubsubsectionStyle[]), 
                "\"Subsubsubsection\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTConvertToSubsubsubsectionStyle[]), 
                Delimiter, 
                "\"Text\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTConvertToTextStyle[]), "\"CodeText\"" :> 
                Module[{$CellContext`bn = ButtonNotebook[]}, 
                  If[Developer`CellInformation[$CellContext`bn] =!= $Failed, 
                    While[
                    FreeQ[
                    Developer`CellInformation[$CellContext`bn], 
                    "CellBracket"], 
                    FrontEndExecute[
                    
                    FrontEnd`SelectionMove[$CellContext`bn, All, Cell, 
                    AutoScroll -> False]]]; 
                    SelectionMove[$CellContext`bn, After, Cell]]; 
                  NotebookWrite[$CellContext`bn, 
                    Cell["XXXX", "CodeText"], All]; 
                  SelectionMove[$CellContext`bn, All, CellContents]], 
                "\"Hyperlink\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTGenerateHyperlinkDialog[]), 
                "\"Display Formula\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertDisplayFormulaCell[]), Delimiter, 
                "\"Solved Example\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`\
WBTInsertSolvedExampleAndSolvedExampleNoteCells[]), 
                "\"Solved Example Input\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertSolvedExampleInputCell[]), 
                "\"Solved Example Text\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertSolvedExampleNoteCell[]), 
                "\"Solved Example Display Formula\"" :> (
                 Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertSolvedExampleDisplayFormulaCell[]),
                 "\"Solved Example Display Formula Numbered\"" :> (
                 Needs["WolframBookTools`"]; 
                 WolframBookTools`\
WBTInsertSolvedExampleDisplayFormulaNumberedCell[]), Delimiter, 
                "\"Theorem\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertTheoremCell[]), 
                "\"Theorem Numbered\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertTheoremNumberedCell[]), 
                "\"Theorem Text\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertTheoremTextCell[]), 
                "\"Proof/Theorem Display Formula\"" :> (
                 Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertProofTheoremDisplayCell[]), 
                "\"Proof/Theorem Display Formula Numbered\"" :> (
                 Needs["WolframBookTools`"]; 
                 WolframBookTools`\
WBTInsertProofTheoremDisplayFormulaNumberedCell[]), 
                "\"Proof\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertProofCell[]), 
                "\"Proof Text\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertProofTextCell[]), Delimiter, 
                "\"Vocabulary 2Col\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertVocabularySectionAnd2ColTable[]), 
                "\"Vocabulary 3Col\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertVocabularySectionAnd3ColTable[]), 
                "\"Add Row\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTVocabTableAddRow[]), 
                "\"Remove Row\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTVocabTableAddRow["RemoveRow" -> True]), 
                "\"Apply Divider\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTVocabTableAddRow["AddDivider" -> True]), 
                "\"Remove Divider\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTVocabTableAddRow[
                  "RemoveDivider" -> True]), Delimiter, 
                "\"Exercises\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertExerciseSection[]), 
                "\"Exercise\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertExerciseSection[]), 
                "\"Exercise Note\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertExerciseNote[]), 
                "\"Solution Section\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertExerciseSolution[]), 
                "\"Solution Text\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertSolutionAnswer[]), Delimiter, 
                "\"Q&A | Tech Notes | More\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertQAndATechNotesMore[]), 
                "\"Question\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertQuestion[]), 
                "\"Answer\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertAnswer[]), 
                "\"Tech Note\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertTechNote[]), 
                "\"More to Explore\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertMoreExplore[]), Delimiter, 
                "\"References Section\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTInsertReferencesSection[]), 
                "\"Add Reference\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTAddReferenceCell[]), Delimiter, 
                "\"Comment\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTApplyComment[])}, Appearance -> 
               "PopupMenu", FrameMargins -> 2.4], 
              ItemBox["", ItemSize -> Fit], 
              ActionMenuBox[
              "\"Function Name\"", {
               "\"Link\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTMakeSymbolLink[]), 
                "\"Format\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTFormatSymbol[])}, Appearance -> 
               "PopupMenu", Method -> "Queued", FrameMargins -> 2.4], 
              ButtonBox[
              "\"Resource Function Link\"", 
               ButtonFunction :> (Needs["WolframBookTools`"]; 
                WolframBookTools`WBTResourceFunctionLink[]), Evaluator -> 
               Automatic, Appearance -> "DialogBox", Method -> "Queued"], 
              ActionMenuBox[
              "\"Inline Format\"", {
               "\"Inline Code\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTConvertToInlineCode[]), 
                "\"Traditional Math\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTTraditionalFormCell[])}, Appearance -> 
               "PopupMenu", Method -> "Queued", FrameMargins -> 2.4], 
              ActionMenuBox[
              "\"Mark Cells\"", {
               "\"Tentative\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTApplyTentative[]), Delimiter, 
                "\"Print Only\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTApplyPrintOnly[]), 
                "\"Web Only\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTApplyWebOnly[]), Delimiter, 
                "\"Locked\"" :> (Needs["WolframBookTools`"]; 
                 WolframBookTools`WBTApplyLock[])}, Method -> "Queued", 
               FrameMargins -> 2.4], 
              ItemBox["", ItemSize -> Fit], 
              GridBox[{{
                 TooltipBox[
                  ButtonBox[
                  "\"Index Field\"", Appearance -> {"AbuttingRight"}, 
                   ButtonFunction :> (Needs["WolframBookTools`"]; 
                    WolframBookTools`WBTInsertIndexField[]), Evaluator -> 
                   Automatic, Method -> "Queued"], 
                  "First put the cursor in a cell to insert a corresponding \
field above to insert index entries.\nList main index entries corresponding \
to the cell separated by commas. Any associated sub-entries\nshould be put \
immediately after in braces.\nExample: programming{functional, rule based}, \
function, relation, lists{flat, nested}\n If $Language is \"ChineseSimplified\
\", \"ChineseTraditional\", \"Japanese\", \"Korean\" or \"Russian\"\ntwo \
index fields get inserted above a selected cell, the top one for inserting \
entries\nintended for the Englsh index and the bottom for the index in the \
language given by $Language.", TooltipDelay -> 0.3], 
                 TooltipBox[
                  ButtonBox[
                  "\"Toggle\"", Appearance -> {"AbuttingLeft"}, 
                   ButtonFunction :> (CurrentValue[
                    ButtonNotebook[], ScreenStyleEnvironment] = 
                    If[CurrentValue[
                    ButtonNotebook[], ScreenStyleEnvironment] === 
                    "Invisibles", "Working", "Invisibles"]), Evaluator -> 
                   Automatic, Method -> "Queued"], 
                  "Toggles index entry fields.", TooltipDelay -> 0.3]}}, 
               GridBoxItemSize -> {
                "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, 
               GridBoxSpacings -> {
                "ColumnsIndexed" -> {2 -> 0}, "Rows" -> {{Automatic}}}], 
              InterpretationBox[
               StyleBox[
                
                GraphicsBox[{}, BaselinePosition -> Baseline, 
                 ImageSize -> {4, 0}], CacheGraphics -> False], 
               Spacer[4]]}}]], "DockedCell", 
          CellFrameMargins -> {{0, 0}, {8, 8}}]}, CreateCellID -> True, 
       TrackCellChangeTimes -> False, ShowPredictiveInterface -> False, 
       AutoStyleOptions -> {
        "UndefinedSymbolStyle" -> 
         None}, $CellContext`GhostCellInEmptyNotebook -> False]}, Closed]], 
   Cell[
    CellGroupData[{
      Cell["Styles for Headers and Footers", "Section"], 
      Cell[
       StyleData["Header"], 
       PrintingOptions -> {"PageHeaderMargins" -> {28, 28}}, StyleMenuListing -> 
       None, FontFamily -> "Source Sans Pro", FontSize -> 8, FontWeight -> 
       "Regular", FontSlant -> "Plain", 
       PrivateFontOptions -> {"FontPostScriptName" -> "Automatic"}, FontColor -> 
       GrayLevel[0.45]], 
      Cell[
       StyleData["PageNumber"], CellMargins -> {{0, 0}, {4, 1}}, 
       StyleMenuListing -> None, FontFamily -> "Source Sans Pro", FontSize -> 
       8, FontWeight -> "Bold", FontSlant -> "Plain", 
       PrivateFontOptions -> {"FontPostScriptName" -> "Automatic"}, FontColor -> 
       GrayLevel[0]]}, Closed]], 
   Cell[
    CellGroupData[{
      Cell[
      "Custom Styles", "Section", 
       CellChangeTimes -> {{3.646611660445352*^9, 3.646611669493331*^9}}], 
      Cell[
       StyleData["Notebook"], WindowToolbars -> {}, PageWidth :> WindowWidth, 
       PrintingOptions -> {
        "FirstPageHeader" -> True, "InnerOuterMargins" -> {30, 30}, 
         "PaperOrientation" -> "Portrait", "PaperSize" -> {504, 720}, 
         "PrintingMargins" -> {{20, 20}, {54, 54}}}, CellLabelAutoDelete -> 
       False, ShowSyntaxStyles -> True, 
       LightDarkAutoColorRules -> <|"AllColors" -> True|>, MenuSortingValue -> 
       None, GraphicsBoxOptions -> {ImageSize -> 240}, 
       Graphics3DBoxOptions -> {ImageSize -> 240}], 
      Cell[
       StyleData["TOCPage"], CellSize -> {40, Inherited}, TextAlignment -> 
       Right, MenuSortingValue -> 20010, StyleMenuListing -> None, FontFamily -> 
       "Source Sans Pro", FontSize -> 18, FontWeight -> "Plain"], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ContentsTitle"], CellMargins -> {{66, 66}, {4, 14}}, 
          MenuSortingValue -> 20000, FontFamily -> "Helvetica", FontSize -> 
          36, FontWeight -> "Bold", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
         Cell[
          StyleData["ContentsTitle", "Printout"]]}, Open]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["TOCTitle"], CellMargins -> {{66, 66}, {4, 14}}, 
          CellGroupingRules -> {"SectionGrouping", 70}, TabSpacings -> 36, 
          CounterIncrements -> "TOCTitle", MenuSortingValue -> 20020, 
          FontFamily -> "Source Sans Pro", FontSize -> 18, FontWeight -> 
          "Bold", FontColor -> GrayLevel[0]], 
         Cell[
          StyleData["TOCTitle", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["TOCSection"], CellMargins -> {{95, 95}, {1, 1}}, 
          CellGroupingRules -> {"SectionGrouping", 75}, TabSpacings -> 34, 
          CounterIncrements -> "TOCSection", MenuSortingValue -> 20030, 
          FontFamily -> "Source Sans Pro", FontSize -> 16], 
         Cell[
          StyleData["TOCSection", "Printout"]]}, Closed]], 
      Cell[
       StyleData["Leader"], TabFilling -> ".", MenuSortingValue -> 10000, 
       StyleMenuListing -> None, FontFamily -> "Source Sans Pro", FontSize -> 
       18, FontWeight -> "Plain"], 
      Cell[
       StyleData["LeaderNoDots"], TabFilling -> " ", MenuSortingValue -> 
       10000, StyleMenuListing -> None, FontFamily -> "Source Sans Pro", 
       FontSize -> 18, FontWeight -> "Plain"], 
      Cell[
       StyleData["PageLink"], MenuSortingValue -> 10000, StyleMenuListing -> 
       None, ButtonStyleMenuListing -> Automatic, FontFamily -> 
       "Source Sans Pro", FontSize -> 18, FontColor -> 
       RGBColor[0.269993, 0.308507, 0.6], 
       ButtonBoxOptions -> {
        Active -> True, Appearance -> {Automatic, None}, 
         ButtonFunction :> (FrontEndExecute[{
            FrontEnd`NotebookLocate[#2]}]& )}], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Title"], CellMargins -> {{64, Inherited}, {50, 72}}, 
          MenuCommandKey :> None, FontFamily -> "Verdana", FontSize -> 42, 
          FontWeight -> "Bold", FontColor -> GrayLevel[0.4]], 
         Cell[
          StyleData["Title", "Printout"], 
          CellMargins -> {{64, Inherited}, {50, 72}}]}, Closed]], 
      Cell[
       StyleData["Chapter"], MenuCommandKey :> None], 
      Cell[
       StyleData["Subchapter"], MenuCommandKey :> None], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Section"], CellFrame -> {{0, 0}, {0, 0}}, 
          CellMargins -> {{64, Inherited}, {12, 64}}, 
          CellGroupingRules -> {"SectionGrouping", 30}, PageBreakAbove -> 
          True, ShowCellLabel -> False, 
          CellFrameLabelMargins -> {{2, Inherited}, {Inherited, Inherited}}, 
          CellChangeTimes -> {3.652728456208679*^9, 3.652728527108994*^9}, 
          CounterIncrements -> "Section", 
          CounterAssignments -> {{"Subsection", 0}, {"Subsubsection", 0}, {
            "Exercise", 0}}, FontFamily -> "Source Sans Pro Semibold", 
          FontSize -> 26, FontSlant -> "Plain", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          InterpretationBoxOptions -> {Editable -> True}], 
         Cell[
          StyleData["Section", "Printout"], 
          CellMargins -> {{62, 4}, {52, 182}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SectionSingleDigit"], CellFrame -> {{0, 0}, {0, 0}}, 
          CellDingbat -> Cell[
            TextData[
             CounterBox["Section"], 
             StyleBox[
             "|\[NegativeThinSpace]\[VeryThinSpace]", FontColor -> 
              RGBColor[
               0.9019607843137255, 0.4117647058823529, 0.027450980392156862`],
               FontFamily -> "Courier", FontSize -> 36, FontWeight -> Plain]],
             "SectionNumber"], CellMargins -> {{100, 100}, {12, 64}}, 
          Evaluatable -> True, CellGroupingRules -> {"SectionGrouping", 60}, 
          PageBreakAbove -> True, ShowCellLabel -> False, 
          CellFrameLabelMargins -> {{2, Inherited}, {Inherited, Inherited}}, 
          CellChangeTimes -> {3.652728456208679*^9, 3.652728527108994*^9}, 
          CounterIncrements -> "Section", 
          CounterAssignments -> {{"Subsection", 0}, {"Subsubsection", 0}, {
            "Exercise", 0}}, FontFamily -> "Source Sans Pro Semibold", 
          FontSize -> 26, FontSlant -> "Plain", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          InterpretationBoxOptions -> {Editable -> True}], 
         Cell[
          StyleData["SectionSingleDigit", "Printout"], 
          CellMargins -> {{100, Inherited}, {12, 2}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SectionDoubleDigit"], CellFrame -> {{0, 0}, {0, 0}}, 
          CellDingbat -> Cell[
            TextData[
             CounterBox["Section"], 
             StyleBox[
             "|\[NegativeThinSpace]\[VeryThinSpace]", FontColor -> 
              RGBColor[
               0.9019607843137255, 0.4117647058823529, 0.027450980392156862`],
               FontFamily -> "Courier", FontSize -> 36, FontWeight -> Plain]],
             "SectionNumber"], CellMargins -> {{112, Inherited}, {12, 64}}, 
          Evaluatable -> True, CellGroupingRules -> {"SectionGrouping", 60}, 
          PageBreakAbove -> True, ShowCellLabel -> False, 
          CellFrameLabelMargins -> {{2, Inherited}, {Inherited, Inherited}}, 
          CellChangeTimes -> {3.652728456208679*^9, 3.652728527108994*^9}, 
          CounterIncrements -> "Section", 
          CounterAssignments -> {{"Subsection", 0}, {"Subsubsection", 0}, {
            "Exercise", 0}}, FontFamily -> "Source Sans Pro Semibold", 
          FontSize -> 26, FontSlant -> "Plain", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          InterpretationBoxOptions -> {Editable -> True}], 
         Cell[
          StyleData["SectionDoubleDigit", "Printout"], 
          CellMargins -> {{112, Inherited}, {12, 2}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SectionNumber"], CellBaseline -> Baseline, FontFamily -> 
          "Source Sans Pro Semibold", FontSize -> 26, FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
         Cell[
          StyleData["SectionNumber", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SectionBar"], FontFamily -> "Source Sans Pro", 
          FontWeight -> "Plain", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
         Cell[
          StyleData["SectionBar", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SectionInline"], FontFamily -> 
          "Source Sans Pro Semibold", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
         Cell[
          StyleData["SectionInline", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SectionNoDingbat"], 
          CellMargins -> {{64, Inherited}, {12, 64}}, Evaluatable -> True, 
          CellGroupingRules -> {"SectionGrouping", 60}, PageBreakAbove -> 
          True, ShowCellLabel -> False, 
          CellFrameLabelMargins -> {{2, Inherited}, {Inherited, Inherited}}, 
          CellChangeTimes -> {3.652728456208679*^9, 3.652728527108994*^9}, 
          CounterIncrements -> "Section", 
          CounterAssignments -> {{"Subsection", 0}, {"Subsubsection", 0}, {
            "Exercise", 0}}, FontFamily -> "Source Sans Pro Semibold", 
          FontSize -> 26, FontSlant -> "Plain", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          InterpretationBoxOptions -> {Editable -> True}], 
         Cell[
          StyleData["SectionNoDingbat", "Printout"], 
          CellMargins -> {{64, Inherited}, {12, 2}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SubsectionNumbered"], CellDingbat -> Cell[
            TextData[{"  ", 
              CounterBox["Section"], ".", 
              CounterBox["Subsection"]}], "SubsectionNumber", FontColor -> 
            RGBColor[
             0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
          CellMargins -> {{98, Inherited}, {Inherited, 32}}, 
          CounterIncrements -> "Subsection", FontFamily -> 
          "Source Sans Pro Semibold", FontSize -> 18, FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
         Cell[
          StyleData["SubsectionNumbered", "Printout"], 
          CellMargins -> {{102, Inherited}, {12, 32}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SubsectionNumber"], FontFamily -> 
          "Source Sans Pro Semibold", FontSize -> 18, FontColor -> 
          GrayLevel[0.25]], 
         Cell[
          StyleData["SubsectionNumber", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SubsectionFirst"], 
          CellMargins -> {{64, Inherited}, {2, 72}}, FontFamily -> 
          "Source Sans Pro Semibold", FontSize -> 18, FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
         Cell[
          StyleData["Subsection", "Printout"], 
          CellMargins -> {{64, Inherited}, {12, 32}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Subsection"], CellMargins -> {{64, Inherited}, {2, 32}}, 
          FontFamily -> "Source Sans Pro Semibold", FontSize -> 18, FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
         Cell[
          StyleData["Subsection", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Subsubsection"], 
          CellMargins -> {{64, Inherited}, {2, 32}}, FontFamily -> 
          "Source Sans Pro Semibold", FontSize -> 16, FontSlant -> "Italic", 
          FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
         Cell[
          StyleData["Subsubsection", "Printout"], 
          CellMargins -> {{64, Inherited}, {6, 14}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SubsubsectionTopMargin"], 
          CellMargins -> {{64, Inherited}, {2, 14}}, FontFamily -> 
          "Source Sans Pro Semibold", FontSize -> 16, FontSlant -> "Italic", 
          FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
         Cell[
          StyleData["SubsubsectionTopMargin", "Printout"], 
          CellMargins -> {{64, Inherited}, {6, 14}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Subsubsubsection"], 
          CellMargins -> {{64, Inherited}, {2, 32}}, FontFamily -> 
          "Source Sans Pro Semibold", FontSize -> 15, FontSlant -> "Plain", 
          FontColor -> GrayLevel[0.45]], 
         Cell[
          StyleData["Subsubsubsection", "Printout"], 
          CellMargins -> {{64, Inherited}, {6, 14}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Subsubsubsubsection"], 
          CellMargins -> {{66, Inherited}, {2, 10}}, 
          StyleKeyMapping -> {
           "Backspace" -> 
            "Subsubsubsection", $CellContext`KeyEvent[
             "Tab", $CellContext`Modifiers -> {"Shift"}] -> 
            "Subsubsubsection"}, CellGroupingRules -> {"SectionGrouping", 70},
           PageBreakBelow -> False, LanguageCategory -> "NaturalLanguage", 
          CounterIncrements -> "Subsubsubsubsection", 
          CounterAssignments -> {{"Item", 0}, {"Subitem", 0}, {
            "Subsubitem", 0}, {"ItemNumbered", 0}, {"SubitemNumbered", 0}, {
            "SubsubitemNumbered", 0}}, MenuSortingValue -> None, FontFamily -> 
          "Source Sans Pro Semibold", FontSize -> 15, FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
         Cell[
          StyleData["Subsubsubsubsection", "Printout"], 
          CellMargins -> {{66, 66}, {2, 10}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ResourcesSubsection"], 
          CellMargins -> {{64, Inherited}, {4, 16}}, FontFamily -> 
          "Source Sans Pro Semibold", FontSize -> 15, FontSlant -> "Plain", 
          FontColor -> GrayLevel[0]], 
         Cell[
          StyleData["ResourcesSubsection", "Printout"], 
          CellMargins -> {{64, Inherited}, {4, 16}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["VocabularySection"], CellFrame -> {{0, 0}, {0, 3}}, 
          CellMargins -> {{64, Inherited}, {8, 30}}, 
          CellGroupingRules -> {"SectionGrouping", 60}, PageBreakBelow -> 
          False, CellFrameMargins -> {{8, 8}, {4, 6}}, CellFrameColor -> 
          RGBColor[0.901961, 0.411765, 0.027451], FontFamily -> 
          "Source Sans Pro", FontSize -> 18, FontSlant -> "Italic", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          GridBoxOptions -> {GridBoxAlignment -> {}}], 
         Cell[
          StyleData["VocabularySection", "Printout"], 
          CellMargins -> {{64, Inherited}, {8, 24}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["VocabularySubsection"], 
          CellMargins -> {{64, Inherited}, {2, 10}}, FontFamily -> 
          "Source Sans Pro", FontSize -> 16, FontSlant -> "Italic", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
         Cell[
          StyleData["VocabularySubsubsection", "Printout"], 
          CellMargins -> {{64, Inherited}, {6, 14}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["VocabularySubsectionSM"], 
          CellFrame -> {{0, 0}, {0.75, 0}}, 
          CellMargins -> {{64, Inherited}, {5, 15}}, 
          CellFrameMargins -> {{8, 8}, {5, 6}}, CellFrameColor -> 
          RGBColor[0.901961, 0.411765, 0.027451], FontFamily -> 
          "Source Sans Pro", FontSize -> 14, FontWeight -> "Bold", FontSlant -> 
          "Regular", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
         Cell[
          StyleData["VocabularySubsubsectionSM", "Printout"], 
          CellMargins -> {{64, Inherited}, {6, 15}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["VocabularyTable", StyleDefinitions -> StyleData["Text"]],
           SpanMaxSize -> 1, FontFamily -> "Source Sans Pro Light", FontColor -> 
          GrayLevel[0.8], 
          GridBoxOptions -> {
           GridBoxAlignment -> {"Columns" -> {{Left}}}, 
            GridBoxDividers -> {"Columns" -> {{False}}, "Rows" -> {False, {
                 Directive[
                  Thickness[0.5], "VocabDivColor"]}, False}}, 
            GridBoxItemStyle -> {
             "Columns" -> {"VocabularyCode", Inherited, Inherited}, 
              "Rows" -> {{Automatic}}}, 
            GridBoxSpacings -> {
             "Columns" -> {{0, 1, 2, 1, 2}}, "Rows" -> {{1}}}}], 
         Cell[
          StyleData["VocabularyTable", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "VocabularyTable3Column", StyleDefinitions -> StyleData["Text"]], 
          SpanMaxSize -> 1, FontFamily -> "Source Sans Pro Light", FontColor -> 
          GrayLevel[0.8], 
          GridBoxOptions -> {
           GridBoxAlignment -> {"Columns" -> {{Left}}}, 
            GridBoxDividers -> {"Columns" -> {{False}}, "Rows" -> {False, {
                 Directive[
                  Thickness[0.5], "VocabDivColor"]}, False}}, 
            GridBoxItemStyle -> {
             "Columns" -> {
               "VocabularyCode", Inherited, "VocabularyCode", Inherited, 
                Inherited}, "Rows" -> {{Automatic}}}, 
            GridBoxSpacings -> {
             "Columns" -> {{0, 1, 2, 1, 2}}, "Rows" -> {{1}}}}], 
         Cell[
          StyleData["VocabularyTable3Column", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["VocabularyText"], FontFamily -> "Source Sans Pro", 
          FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain", 
          FontColor -> GrayLevel[0.25]], 
         Cell[
          StyleData["VocabularyText", "Printout"], 
          CellMargins -> {{66, 10}, {7, 7}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["VocabularyCode", StyleDefinitions -> StyleData["Text"]], 
          Deletable -> False, FontFamily -> "Source Sans Pro", FontSize -> 14,
           FontWeight -> "Bold", FontColor -> GrayLevel[0.25], 
          GridBoxOptions -> {
           GridBoxAlignment -> {"Columns" -> {{Left}}}, 
            GridBoxDividers -> {
             "Columns" -> {{False}}, "ColumnsIndexed" -> {}, 
              "Rows" -> {True, {True}, True}, "RowsIndexed" -> {}}}], 
         Cell[
          StyleData["VocabularyCode", "Printout"], FontColor -> 
          GrayLevel[0.25]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["GrayPointSix"], FontColor -> GrayLevel[0.6]], 
         Cell[
          StyleData["GrayPointSix", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["VocabDivColor"], FontColor -> GrayLevel[0.8]], 
         Cell[
          StyleData["VocabDivColor", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SummarySection"], PageWidth :> WindowWidth, 
          CellFrame -> {{0, 0}, {0, 3}}, 
          CellMargins -> {{64, Inherited}, {8, 30}}, 
          CellGroupingRules -> {"SectionGrouping", 60}, PageBreakAbove -> 
          True, CellFrameMargins -> {{8, 8}, {4, 6}}, CellFrameColor -> 
          RGBColor[0.901961, 0.411765, 0.027451], MenuSortingValue -> 9000, 
          FontFamily -> "Source Sans Pro", FontSize -> 18, FontSlant -> 
          "Italic", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          GridBoxOptions -> {GridBoxAlignment -> {}}], 
         Cell[
          StyleData["SummarySection", "Printout"], 
          CellMargins -> {{64, 64}, {8, 30}}]}, Open]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SummaryNote"], PageWidth :> WindowWidth, 
          CellMargins -> {{64, 64}, {4, 8}}, ReturnCreatesNewCell -> True, 
          PageBreakBelow -> False, MenuSortingValue -> 9010, FontFamily -> 
          "Source Sans Pro", FontSize -> 14, FontWeight -> "Plain", FontSlant -> 
          "Plain", FontColor -> GrayLevel[0.25]], 
         Cell[
          StyleData["SummaryNote", "Printout"], 
          CellMargins -> {{68, 68}, {4, 8}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SummaryList"], PageWidth :> WindowWidth, CellDingbat -> 
          Cell[
            TextData[{"\[FilledSmallSquare]"}], FontSize -> 12, FontColor -> 
            GrayLevel[0.5]], CellMargins -> {{78, 78}, {4, 8}}, 
          ReturnCreatesNewCell -> True, CellFrameLabelMargins -> 3, 
          LineSpacing -> {1.25, 1., 2.}, MenuSortingValue -> 9020, FontFamily -> 
          "Source Sans Pro", FontSize -> 14, FontWeight -> "Plain", FontSlant -> 
          "Plain", FontColor -> GrayLevel[0.25]], 
         Cell[
          StyleData["SummaryList", "Printout"], 
          CellMargins -> {{82, 82}, {4, 8}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ExerciseSection"], CellFrame -> {{0, 0}, {0, 3}}, 
          CellMargins -> {{64, Inherited}, {8, 30}}, Evaluatable -> True, 
          CellGroupingRules -> {"SectionGrouping", 60}, PageBreakBelow -> 
          False, ShowCellLabel -> False, CellFrameMargins -> {{8, 8}, {4, 6}},
           CellFrameColor -> RGBColor[0.901961, 0.411765, 0.027451], 
          CounterAssignments -> {{"Subsection", 0}, {"Subsubsection", 0}, {
            "Exercise", 0}}, FontFamily -> "Source Sans Pro", FontSize -> 18, 
          FontSlant -> "Italic", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          GridBoxOptions -> {GridBoxAlignment -> {}}], 
         Cell[
          StyleData["ExerciseSection", "Printout"], 
          CellMargins -> {{64, Inherited}, {8, 30}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ExerciseSubsection"], PageWidth :> 500, 
          CellFrame -> {{False, False}, {1, False}}, 
          CellMargins -> {{64, Inherited}, {8, 18}}, 
          CellGroupingRules -> {"SectionGrouping", 61}, CellFrameColor -> 
          RGBColor[0.65, 0.6, 0.6], FontFamily -> "Source Sans Pro", FontSize -> 
          18, FontSlant -> "Italic", FontColor -> RGBColor[0.65, 0.6, 0.6]], 
         Cell[
          StyleData["ExerciseSubsection", "Printout"], 
          CellFrame -> {{False, False}, {1, False}}, CellFrameColor -> 
          RGBColor[0.65, 0.6, 0.6]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ExerciseSubsectionSM"], 
          CellMargins -> {{64, Inherited}, {4, 12}}, 
          CellGroupingRules -> {"SectionGrouping", 61}, FontFamily -> 
          "Source Sans Pro Semibold", FontSize -> 15, FontSlant -> "Plain", 
          FontColor -> GrayLevel[0]], 
         Cell[
          StyleData["ExerciseSubsectionSM", "Printout"], 
          CellMargins -> {{64, Inherited}, {4, 12}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Exercise"], CellDingbat -> Cell[
            TextData[{"  ", 
              CounterBox["Section"], ".", 
              CounterBox["Exercise"]}], "ExerciseNumber", FontColor -> 
            RGBColor[
             0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
          ShowGroupOpener -> False, CellMargins -> {{106, Inherited}, {4, 8}},
           CellGroupingRules -> {"SectionGrouping", 62}, DefaultNewCellStyle -> 
          "ExerciseInput", CounterIncrements -> "Exercise", MenuSortingValue -> 
          10020, MenuCommandKey :> "2", FontFamily -> "Source Sans Pro", 
          FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain", 
          FontColor -> GrayLevel[0.25]], 
         Cell[
          StyleData["Exercise", "Printout"], ShowGroupOpener -> False, 
          CellMargins -> {{108, Inherited}, {4, 8}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ExerciseNumber"], FontFamily -> "Source Sans Pro", 
          FontSize -> 14, FontWeight -> "Bold", FontColor -> GrayLevel[0.25]], 
         Cell[
          StyleData["ExerciseNumber", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ExerciseSectionNote"], 
          CellMargins -> {{64, Inherited}, {4, 8}}, ReturnCreatesNewCell -> 
          True, CellGroupingRules -> {"SectionGrouping", 62.}, PageBreakBelow -> 
          False, LineSpacing -> {1., 1., 2.}, MenuSortingValue -> 10050, 
          FontFamily -> "Source Sans Pro", FontSize -> 14, FontWeight -> 
          "Plain", FontSlant -> "Plain", FontColor -> GrayLevel[0.25]], 
         Cell[
          StyleData["ExerciseSectionNote", "Printout"], 
          CellMargins -> {{64, Inherited}, {4, 8}}, 
          LineSpacing -> {0.8, 1., 2.}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ExerciseNote"], 
          CellMargins -> {{106, Inherited}, {4, 8}}, ReturnCreatesNewCell -> 
          True, CellGroupingRules -> {"SectionGrouping", 62.}, PageBreakBelow -> 
          False, LineSpacing -> {1., 1., 2.}, MenuSortingValue -> 10050, 
          FontFamily -> "Source Sans Pro", FontSize -> 14, FontWeight -> 
          "Plain", FontSlant -> "Plain", FontColor -> GrayLevel[0.25]], 
         Cell[
          StyleData["ExerciseNote", "Printout"], 
          LineSpacing -> {0.8, 1., 2.}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ExerciseSolution"], ShowGroupOpener -> "Inline", 
          CellMargins -> {{106, Inherited}, {4, 8}}, ReturnCreatesNewCell -> 
          True, CellGroupingRules -> {"SectionGrouping", 62.}, PageBreakBelow -> 
          False, CellChangeTimes -> {3.938884624675207*^9}, 
          LineSpacing -> {1., 1., 2.}, 
          CounterAssignments -> {{"ItemNumbered", 0}}, LineColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          FrontFaceColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          BackFaceColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          GraphicsColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          MenuSortingValue -> 10050, FontFamily -> "Source Sans Pro Semibold",
           FontSize -> 14, FontSlant -> "Plain", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
         Cell[
          StyleData["ExerciseSolution", "Printout"], 
          CellMargins -> {{106, Inherited}, {4, 8}}, 
          LineSpacing -> {0.8, 1., 2.}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SolutionAnswer"], CellMargins -> {{106, 10}, {4, 1}}, 
          ReturnCreatesNewCell -> True, LineSpacing -> {1.125, 1., 2.}, 
          MenuSortingValue -> 10032, FontFamily -> "Source Sans Pro", 
          FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain", 
          FontColor -> 
          GrayLevel[0.25], $CellContext`DefaultReturnCreatedCellStyle -> 
          "Question"], 
         Cell[
          StyleData["SolutionAnswer", "Printout"], 
          CellMargins -> {{106, 10}, {4, 1}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SolutionItem"], CellDingbat -> Cell[
            TextData[{
              CounterBox["ItemNumbered"], ")"}], FontSize -> 14, FontWeight -> 
            "Regular", FontColor -> GrayLevel[0.25]], 
          CellMargins -> {{122, Inherited}, {2, 4}}, ReturnCreatesNewCell -> 
          True, CellFrameLabelMargins -> 4, LineSpacing -> {1.25, 1., 2.}, 
          CounterIncrements -> "ItemNumbered", MenuSortingValue -> 10040, 
          FontFamily -> "Source Sans Pro", FontSize -> 14, FontWeight -> 
          "Plain", FontSlant -> "Plain", FontColor -> GrayLevel[0.25]], 
         Cell[
          StyleData["SolutionItem", "Printout"], 
          CellMargins -> {{106, Inherited}, {2, 4}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["QASection"], CellFrame -> {{0, 0}, {0, 3}}, 
          CellMargins -> {{64, Inherited}, {8, 30}}, 
          CellGroupingRules -> {"SectionGrouping", 60}, PageBreakBelow -> 
          False, CellFrameMargins -> {{8, 8}, {4, 6}}, CellFrameColor -> 
          RGBColor[0.901961, 0.411765, 0.027451], 
          CounterAssignments -> {{"Subsection", 0}, {"Subsubsection", 0}, {
            "Question", 0}}, FontFamily -> "Source Sans Pro", FontSize -> 18, 
          FontSlant -> "Italic", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          GridBoxOptions -> {GridBoxAlignment -> {}}], 
         Cell[
          StyleData["QASection", "Printout"], 
          CellMargins -> {{64, Inherited}, {8, 30}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Question"], CellMargins -> {{64, Inherited}, {2, 12}}, 
          ReturnCreatesNewCell -> True, PageBreakBelow -> False, 
          CounterIncrements -> "Question", MenuSortingValue -> 10030, 
          MenuCommandKey :> "3", FontFamily -> "Source Sans Pro", FontSize -> 
          14, FontWeight -> "Bold", FontSlant -> 
          "Plain", $CellContext`DefaultReturnCreatedCellStyle -> "Answer"], 
         Cell[
          StyleData["Question", "Printout"], 
          CellMargins -> {{65, Inherited}, {2, 12}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Answer"], CellMargins -> {{66, 10}, {4, 1}}, 
          ReturnCreatesNewCell -> True, LineSpacing -> {1.125, 1., 2.}, 
          MenuSortingValue -> 10032, FontFamily -> "Source Sans Pro", 
          FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain", 
          FontColor -> 
          GrayLevel[0.25], $CellContext`DefaultReturnCreatedCellStyle -> 
          "Question"], 
         Cell[
          StyleData["Answer", "Printout"], 
          CellMargins -> {{66, 10}, {4, 1}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["QuestionNumber"], FontFamily -> "Source Sans Pro", 
          FontSize -> 14, FontWeight -> "Bold", FontColor -> GrayLevel[0.25]], 
         Cell[
          StyleData["QuestionNumber", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["TechNoteSection"], CellFrame -> {{0, 0}, {0, 3}}, 
          CellMargins -> {{64, Inherited}, {8, 30}}, 
          CellGroupingRules -> {"SectionGrouping", 60}, PageBreakBelow -> 
          False, CellFrameMargins -> {{8, 8}, {4, 6}}, CellFrameColor -> 
          RGBColor[0.901961, 0.411765, 0.027451], FontFamily -> 
          "Source Sans Pro", FontSize -> 18, FontSlant -> "Italic", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          GridBoxOptions -> {GridBoxAlignment -> {}}], 
         Cell[
          StyleData["TechNoteSection", "Printout"], 
          CellMargins -> {{64, Inherited}, {8, 30}}]}, Open]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["TechNoteSubsection"], 
          CellMargins -> {{64, Inherited}, {2, 10}}, FontFamily -> 
          "Source Sans Pro", FontSize -> 16, FontSlant -> "Italic", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
         Cell[
          StyleData["TechNoteSubsubsection", "Printout"], 
          CellMargins -> {{64, Inherited}, {6, 14}}]}, Open]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["TechNoteItem"], CellDingbat -> Cell[
            TextData[{
              CounterBox["ItemNumbered"], "."}], FontSize -> 12, FontWeight -> 
            Bold, FontColor -> GrayLevel[0.25]], 
          CellMargins -> {{93, Inherited}, {4, 8}}, ReturnCreatesNewCell -> 
          True, CellFrameLabelMargins -> 4, 
          CellChangeTimes -> {{3.651515120816633*^9, 3.651515125814807*^9}, 
            3.6515155174947033`*^9}, LineSpacing -> {1.25, 1., 2.}, 
          CounterIncrements -> "ItemNumbered", MenuSortingValue -> 10040, 
          FontFamily -> "Source Sans Pro", FontSize -> 14, FontWeight -> 
          "Plain", FontSlant -> "Plain", FontColor -> GrayLevel[0.25]], 
         Cell[
          StyleData["TechNoteItem", "Printout"], 
          CellMargins -> {{93, Inherited}, {4, 8}}]}, Open]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["TechNoteNoDingbat"], 
          CellMargins -> {{82, Inherited}, {4, 8}}, ReturnCreatesNewCell -> 
          True, CellFrameLabelMargins -> 4, LineSpacing -> {1.25, 1., 2.}, 
          MenuSortingValue -> 10040, FontFamily -> "Source Sans Pro", 
          FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain", 
          FontColor -> GrayLevel[0.25]], 
         Cell[
          StyleData["TechNote", "Printout"], 
          CellMargins -> {{82, Inherited}, {4, 8}}]}, Open]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["TechNote"], CellDingbat -> Cell[
            TextData[{"\[FilledSquare]"}], FontSize -> 12, FontColor -> 
            GrayLevel[0.5]], CellMargins -> {{78, Inherited}, {4, 8}}, 
          ReturnCreatesNewCell -> True, CellFrameLabelMargins -> 4, 
          LineSpacing -> {1.25, 1., 2.}, MenuSortingValue -> 10040, 
          FontFamily -> "Source Sans Pro", FontSize -> 14, FontWeight -> 
          "Plain", FontSlant -> "Plain", FontColor -> GrayLevel[0.25]], 
         Cell[
          StyleData["TechNote", "Printout"]]}, Open]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["TechNoteInput", StyleDefinitions -> StyleData["Input"]], 
          CellMargins -> {{99, 10}, {3, 7}}, SpanMaxSize -> 1, FontSize -> 
          14], 
         Cell[
          StyleData["TechNoteInput", "Printout"], 
          CellMargins -> {{99, 10}, {3, 7}}]}, Open]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "TechNoteOutput", StyleDefinitions -> StyleData["Output"]], 
          CellMargins -> {{99, 10}, {14, 7}}, FontSize -> 14], 
         Cell[
          StyleData["TechNoteOutput", "Printout"], 
          CellMargins -> {{99, 10}, {14, 7}}]}, Open]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["MoreExploreSection"], CellFrame -> {{0, 0}, {0, 3}}, 
          CellMargins -> {{64, Inherited}, {8, 30}}, 
          CellGroupingRules -> {"SectionGrouping", 60}, PageBreakBelow -> 
          False, CellFrameMargins -> {{8, 8}, {4, 6}}, CellFrameColor -> 
          RGBColor[0.901961, 0.411765, 0.027451], FontFamily -> 
          "Source Sans Pro", FontSize -> 18, FontSlant -> "Italic", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          GridBoxOptions -> {GridBoxAlignment -> {}}], 
         Cell[
          StyleData["MoreExploreSection", "Printout"], 
          CellMargins -> {{64, Inherited}, {8, 30}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["MoreExplore"], CellMargins -> {{64, Inherited}, {4, 8}}, 
          ReturnCreatesNewCell -> True, PageBreakBelow -> False, 
          MenuSortingValue -> 10050, FontFamily -> "Source Sans Pro", 
          FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain", 
          FontColor -> GrayLevel[0]], 
         Cell[
          StyleData["MoreExplore", "Printout"], 
          CellMargins -> {{64, Inherited}, {4, 8}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["MoreExploreShortURL"], 
          CellMargins -> {{64, Inherited}, {4, 8}}, PageBreakBelow -> False, 
          FontFamily -> "Source Sans Pro Light", FontSize -> 14, FontSlant -> 
          "Plain", FontColor -> GrayLevel[0]], 
         Cell[
          StyleData["MoreExploreShortURL", "Printout"], 
          CellMargins -> {{64, Inherited}, {4, 8}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["TakeawaysSection"], CellFrame -> {{1, 1}, {1, 1}}, 
          CellMargins -> {{52, -12}, {-1, 30}}, 
          CellGroupingRules -> {"SectionGrouping", 60}, PageBreakBelow -> 
          False, CellFrameMargins -> {{16, 8}, {6, 6}}, CellFrameColor -> 
          RGBColor[0.901961, 0.411765, 0.027451], FontFamily -> 
          "Source Sans Pro", FontSize -> 15, FontWeight -> "Bold", FontSlant -> 
          "Plain", 
          PrivateFontOptions -> {"FontPostScriptName" -> "Automatic"}, 
          FontColor -> GrayLevel[1], Background -> 
          RGBColor[0.901961, 0.411765, 0.027451]], 
         Cell[
          StyleData["TakeawaysSection", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["TakeawaysHeader"], 
          CellMargins -> {{64, Inherited}, {5, 15}}, 
          CellFrameMargins -> {{8, 8}, {5, 6}}, CellFrameColor -> 
          RGBColor[0.901961, 0.411765, 0.027451], FontFamily -> 
          "Source Sans Pro", FontSize -> 15, FontWeight -> "SemiBold", 
          FontSlant -> "Regular", FontColor -> GrayLevel[0]], 
         Cell[
          StyleData["TakeawaysHeader", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["TakeawaysTable"], CellFrame -> {{0, 0}, {0, 0}}, 
          CellMargins -> {{52, -12}, {12, -1}}, ReturnCreatesNewCell -> True, 
          CellFrameMargins -> {{18, 12}, {20, 12}}, CellFrameLabelMargins -> 
          6, LineSpacing -> {1.25, 1., 2.}, SpanMaxSize -> 1, FontFamily -> 
          "Source Sans Pro Light", FontColor -> GrayLevel[0.25], Background -> 
          RGBColor[1., 0.9490196078431372, 0.8509803921568627], 
          GridBoxOptions -> {
           GridBoxAlignment -> {"Columns" -> {{Left}}}, 
            GridBoxDividers -> {"Columns" -> {{False}}, "Rows" -> {False}}, 
            GridBoxItemStyle -> {
             "Columns" -> {"TakeawaysDingbat", Inherited}, 
              "Rows" -> {{SpanFromLeft, Automatic}}}, 
            GridBoxSpacings -> {"Columns" -> {{0.5}}, "Rows" -> {{0.75}}}}], 
         Cell[
          StyleData["TakeawaysTable", "Printout"], 
          CellMargins -> {{51.5, -12.5}, {12, 0}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["TakeawaysText"], FontFamily -> "Source Sans Pro", 
          FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain", 
          FontColor -> GrayLevel[0]], 
         Cell[
          StyleData["TakeawaysText", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "TakeawaysDingbat", StyleDefinitions -> StyleData["Text"]], 
          FontFamily -> "Source Sans Pro", FontSize -> 10, FontWeight -> 
          "SemiBold", FontColor -> RGBColor[0.901961, 0.411765, 0.027451], 
          GridBoxOptions -> {
           GridBoxAlignment -> {"Columns" -> {{Left}}}, 
            GridBoxDividers -> {
             "Columns" -> {{False}}, "ColumnsIndexed" -> {}, 
              "Rows" -> {True, {True}, True}, "RowsIndexed" -> {}}}], 
         Cell[
          StyleData["TakeawaysDingbat", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["CodeText", StyleDefinitions -> StyleData["Text"]], 
          CellMargins -> {{66, 10}, {3, 15}}, LineSpacing -> {1, 1, 1}, 
          MenuSortingValue -> 1540, 
          MenuCommandKey :> $CellContext`MenuKey[
           "7", $CellContext`Modifiers -> {"Shift"}], FontFamily -> 
          "Source Sans Pro", FontSize -> 16, FontColor -> GrayLevel[0.25]], 
         Cell[
          StyleData["CodeText", "Printout"], 
          CellMargins -> {{66, 10}, {4, 12}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["CodeCaption"], CellFrame -> {{0, 0}, {0, 0}}, 
          CellMargins -> {{66, 10}, {4, 12}}, PageBreakBelow -> False, 
          CellFrameColor -> GrayLevel[0.4], LineSpacing -> {1, 1, 1}, 
          MenuSortingValue -> 10010, MenuCommandKey :> "1", FontFamily -> 
          "Source Sans Pro", FontSize -> 16, FontColor -> GrayLevel[0.25]], 
         Cell[
          StyleData["CodeCaption", "Printout"], 
          CellMargins -> {{66, 10}, {4, 12}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["DisplayFormula"], CellMargins -> {{99, 15}, {2, 12}}, 
          StripStyleOnPaste -> True, DefaultFormatType -> 
          DefaultInputFormatType, 
          HyphenationOptions -> {"HyphenationCharacter" -> "\[Continuation]"},
           LanguageCategory -> "Formula", ScriptLevel -> 0, 
          SingleLetterItalics -> True, MenuSortingValue -> 1700, FontFamily -> 
          "Baskerville", FontSize -> 16, FontWeight -> "Regular", FontSlant -> 
          "Plain", 
          PrivateFontOptions -> {"FontPostScriptName" -> "Automatic"}, 
          FontColor -> GrayLevel[0.2], 
          UnderoverscriptBoxOptions -> {LimitsPositioning -> True}], 
         Cell[
          StyleData["DisplayFormula", "Printout"], 
          CellMargins -> {{99, Inherited}, {Inherited, Inherited}}]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "DisplayFormulaNumbered", StyleDefinitions -> 
           StyleData["DisplayFormula"]], CellFrameLabels -> {{None, 
             Cell[
              TextData[{"(", 
                CounterBox["DisplayFormulaNumbered"], ")"}], 
              "DisplayFormulaEquationNumber", CellSize -> {36, Inherited}]}, {
            None, None}}, 
          CellFrameLabelMargins -> {{Inherited, 6}, {Inherited, Inherited}}, 
          TextAlignment -> Left, CounterIncrements -> 
          "DisplayFormulaNumbered", MenuSortingValue -> 1710, FontFamily -> 
          "Baskerville", FontSize -> 16, FontWeight -> "Regular", FontSlant -> 
          "Plain", 
          PrivateFontOptions -> {"FontPostScriptName" -> "Automatic"}], 
         Cell[
          StyleData["DisplayFormulaNumbered", "Printout"], 
          CellMargins -> {{99, 19}, {Inherited, Inherited}}, 
          CellFrameMargins -> {{Inherited, Inherited}, {
            Inherited, Inherited}}]}, Open]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Print"], CellMargins -> {{99, 10}, {4, 4}}, FontSize -> 
          15], 
         Cell[
          StyleData["Print", "Presentation"], 
          CellMargins -> {{100, Inherited}, {1.5 Inherited, 1.5 Inherited}}], 
         
         Cell[
          StyleData["Print", "SlideShow"], 
          CellMargins -> {{
            0.135 FrontEnd`AbsoluteCurrentValue[{WindowSize, 1}], 0.01 
             FrontEnd`AbsoluteCurrentValue[{WindowSize, 1}]}, {8, 15}}, 
          FontSize -> 18], 
         Cell[
          StyleData["Print", "Condensed"], 
          CellMargins -> {{18, Inherited}, {Inherited, Inherited}}], 
         Cell[
          StyleData["Print", "Printout"], 
          CellMargins -> {{49, Inherited}, {Inherited, Inherited}}], 
         Cell[
          StyleData["Print", "PrintoutGray"], 
          CellMargins -> {{49, Inherited}, {Inherited, Inherited}}]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Item"], CellDingbat -> 
          StyleBox[
           "\:203a", Alignment -> Baseline, FontFamily -> Automatic, FontSize -> 
            18, FontWeight -> GrayLevel[0], 
            RGBColor[
            0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
          CellMargins -> {{81, 10}, {4, 8}}, ReturnCreatesNewCell -> True, 
          StyleKeyMapping -> {"Tab" -> "Subitem", "*" -> "Subitem"}, 
          CellGroupingRules -> {"GroupTogetherNestedGrouping", 15000}, 
          CellFrameLabelMargins -> 4, CounterIncrements -> "Item", 
          CounterAssignments -> {{"Subitem", 0}, {"Subsubitem", 0}}, 
          MenuSortingValue -> 1600, FontFamily -> "Source Sans Pro", FontSize -> 
          15], 
         Cell[
          StyleData["Item", "Printout"], CellDingbat -> 
          StyleBox[
           "\:203a", Alignment -> Baseline, FontSize -> 17, FontFamily -> 
            "Automatic", FontWeight -> "Black", 
            RGBColor[
            0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
          CellMargins -> {{80, Inherited}, {Inherited, Inherited}}, 
          CellDingbatMargin -> 2, 
          CellGroupingRules -> {
           "GroupTogetherNestedGrouping", 1.073741821*^9}, 
          CellChangeTimes -> {{3.8855684925344043`*^9, 
           3.885568495926271*^9}, {3.885643888233819*^9, 
           3.885643900721007*^9}}]}, Open]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ItemParagraph"], CellMargins -> {{81, 10}, {4, 1}}, 
          ReturnCreatesNewCell -> True, 
          StyleKeyMapping -> {"Tab" -> "SubitemParagraph"}, 
          CellGroupingRules -> {"GroupTogetherNestedGrouping", 15000}, 
          MenuSortingValue -> 1600, FontFamily -> "Source Sans Pro", FontSize -> 
          15, FontColor -> GrayLevel[0.2]], 
         Cell[
          StyleData["ItemParagraph", "Printout"], 
          CellMargins -> {{80, Inherited}, {Inherited, 0.5 Inherited}}]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ItemNumbered"], CellDingbat -> Cell[
            TextData[{
              CounterBox["ItemNumbered"], "."}], FontWeight -> "Bold"], 
          CellMargins -> {{85, 10}, {4, 8}}, ReturnCreatesNewCell -> True, 
          StyleKeyMapping -> {"Tab" -> "SubitemNumbered"}, 
          CellGroupingRules -> {"GroupTogetherNestedGrouping", 15000}, 
          CellFrameLabelMargins -> 4, CounterIncrements -> "ItemNumbered", 
          CounterAssignments -> {{"SubitemNumbered", 0}, {
            "SubsubitemNumbered", 0}}, MenuSortingValue -> 1630, FontFamily -> 
          "Source Sans Pro", FontSize -> 15], 
         Cell[
          StyleData["ItemNumbered", "Printout"], 
          CellMargins -> {{90, Inherited}, {Inherited, Inherited}}]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Subitem"], CellDingbat -> 
          StyleBox[
           "\:203a", Alignment -> Baseline, FontSize -> 18, FontWeight -> 
            GrayLevel[0], 
            RGBColor[
            0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
          CellMargins -> {{105, 12}, {4, 4}}, ReturnCreatesNewCell -> True, 
          StyleKeyMapping -> {
           "Tab" -> "Subsubitem", "*" -> "Subsubitem", "Backspace" -> 
            "Item", $CellContext`KeyEvent[
             "Tab", $CellContext`Modifiers -> {"Shift"}] -> "Item"}, 
          CellGroupingRules -> {"ItemGrouping", 80}, CellFrameLabelMargins -> 
          4, CounterIncrements -> "Subitem", 
          CounterAssignments -> {{"Subsubitem", 0}}, MenuSortingValue -> 1610,
           FontFamily -> "Source Sans Pro", FontSize -> 14], 
         Cell[
          StyleData["Subitem", "Printout"], CellDingbat -> 
          StyleBox[
           "\:203a", Alignment -> Baseline, FontSize -> 17, FontFamily -> 
            "Automatic", FontWeight -> "Black", 
            RGBColor[
            0.9019607843137255, 0.4117647058823529, 0.027450980392156862`]], 
          CellMargins -> {{105, Inherited}, {Inherited, Inherited}}]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Subsubitem"], CellDingbat -> 
          StyleBox[
           "\:203a", Alignment -> Baseline, FontSize -> 18, FontWeight -> 
            GrayLevel[0], 
            RGBColor[0.6, 0.6, 0.6]], CellMargins -> {{129, 12}, {4, 4}}, 
          ReturnCreatesNewCell -> True, 
          StyleKeyMapping -> {
           "Backspace" -> 
            "Subitem", $CellContext`KeyEvent[
             "Tab", $CellContext`Modifiers -> {"Shift"}] -> "Subitem"}, 
          CellGroupingRules -> {"ItemGrouping", 90}, CellFrameLabelMargins -> 
          4, CounterIncrements -> "Subsubitem", MenuSortingValue -> 1620, 
          FontFamily -> "Source Sans Pro", FontSize -> 13], 
         Cell[
          StyleData["Subsubitem", "Printout"], CellDingbat -> 
          StyleBox[
           "\:203a", Alignment -> Baseline, FontSize -> 18, FontWeight -> 
            GrayLevel[0], 
            GrayLevel[0.65]], 
          CellMargins -> {{129, Inherited}, {Inherited, Inherited}}]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Note"], CellFrame -> {{0, 0}, {0, 1}}, 
          CellMargins -> {{66, 0}, {8, 6}}, ReturnCreatesNewCell -> True, 
          StyleKeyMapping -> {"Tab" -> "SubitemParagraph"}, 
          CellGroupingRules -> {"GroupTogetherNestedGrouping", 15000}, 
          CellFrameMargins -> {{14, 14}, {12, 12}}, CellFrameColor -> 
          RGBColor[0.901961, 0.411765, 0.027451], 
          LineSpacing -> {1.2, 1., 2.}, MenuSortingValue -> 1600, FontFamily -> 
          "Source Sans Pro", FontSize -> 15, FontColor -> GrayLevel[0.2], 
          Background -> RGBColor[1., 0.9647058823529412, 0.9411764705882353]], 
         Cell[
          StyleData["Note", "Printout"], CellMargins -> {{66, 0}, {10, 8}}, 
          FontSize -> 14]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Echo"], 
          CellMargins -> {{99, Inherited}, {Inherited, Inherited}}], 
         Cell[
          StyleData["Echo", "Printout"], 
          CellMargins -> {{99, Inherited}, {Inherited, Inherited}}]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["EchoTiming"], 
          CellMargins -> {{99, Inherited}, {Inherited, Inherited}}], 
         Cell[
          StyleData["EchoTiming", "Printout"], 
          CellMargins -> {{99, Inherited}, {Inherited, Inherited}}]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Epigraph"], CellMargins -> {{100, 78}, {6, 24}}, 
          CellFrameLabels -> {{None, None}, {None, None}}, 
          CellFrameLabelMargins -> 8, TextAlignment -> Left, Hyphenation -> 
          True, HyphenationOptions -> {"HyphenationMinLengths" -> {3, 5}}, 
          LineSpacing -> {1, 2}, TabSpacings -> {1.2, 2.5}, FontFamily -> 
          "Source Sans Pro", FontSize -> 18, FontWeight -> "Plain", FontSlant -> 
          "Plain"], 
         Cell[
          StyleData["Epigraph", "Printout", StyleDefinitions -> None]]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["EpigraphAuthor"], CellMargins -> {{125, 100}, {12, 12}}, 
          TextAlignment -> Left, LineSpacing -> {1, 2}, 
          ParagraphIndent -> -17, TabSpacings -> {1.5, 2.5}, FontFamily -> 
          "Source Sans Pro", FontSize -> 16, FontWeight -> "Plain", FontSlant -> 
          "Plain", FontTracking -> -0.05], 
         Cell[
          StyleData["EpigraphAuthor", "Printout", StyleDefinitions -> None]]},
         Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ReferenceSection"], CellFrame -> {{0, 0}, {0, 3}}, 
          CellMargins -> {{64, Inherited}, {8, 30}}, 
          CellGroupingRules -> {"SectionGrouping", 60}, PageBreakBelow -> 
          False, CellFrameMargins -> {{8, 8}, {4, 6}}, CellFrameColor -> 
          RGBColor[0.901961, 0.411765, 0.027451], FontFamily -> 
          "Source Sans Pro", FontSize -> 18, FontSlant -> "Italic", FontColor -> 
          RGBColor[
           0.9019607843137255, 0.4117647058823529, 0.027450980392156862`], 
          GridBoxOptions -> {GridBoxAlignment -> {}}], 
         Cell[
          StyleData["ReferenceSection", "Printout"], 
          CellMargins -> {{64, Inherited}, {8, 30}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Reference"], CellDingbat -> Cell[
            TextData[{
              CounterBox["Reference"]}], "SR", FontSize -> 13, 
            CellSize -> {30, Inherited}, TextAlignment -> Right], 
          CellMargins -> {{80, 10}, {3, 4}}, 
          CellFrameLabels -> {{None, None}, {None, None}}, 
          CellFrameLabelMargins -> 8, 
          CellChangeTimes -> {{3.738599543211059*^9, 3.738599549834622*^9}}, 
          LineSpacing -> {1, 2}, ParagraphIndent -> -8, 
          TabSpacings -> {0.7, 2.5}, CounterIncrements -> "Reference", 
          MenuSortingValue -> 1870, FontFamily -> "Source Sans Pro", FontSize -> 
          14, FontWeight -> "Plain", FontSlant -> "Plain"], 
         Cell[
          StyleData["Reference", "Printout", StyleDefinitions -> None], 
          CellMargins -> {{66, 10}, {3, 4}}, TabSpacings -> {1.2, 2.5}]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["TextFirst"], CellMargins -> {{66, 10}, {4, 72}}, 
          ReturnCreatesNewCell -> True, FontFamily -> "Baskerville", FontSize -> 
          18, FontColor -> GrayLevel[0]], 
         Cell[
          StyleData["Text", "Printout"], CellMargins -> {{66, 10}, {7, 7}}]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Text"], CellMargins -> {{66, 10}, {4, 12}}, 
          ReturnCreatesNewCell -> True, 
          CellChangeTimes -> {{3.968834509197134*^9, 3.968834509700926*^9}}, 
          FontFamily -> "Baskerville", FontSize -> 18, FontColor -> 
          RGBColor[
           0.050002574846150734`, 0.050002574846150734`, 
            0.050002574846150734`]], 
         Cell[
          StyleData["Text", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SmallText"], CellMargins -> {{66, 10}, {6, 6}}, 
          LineSpacing -> {1, 3}, TabSpacings -> {2.5}, LanguageCategory -> 
          "NaturalLanguage", CounterIncrements -> "SmallText", 
          MenuSortingValue -> 1750, StyleMenuListing -> None, FontFamily -> 
          "Source Sans Pro", FontSize -> 14, FontColor -> GrayLevel[0.2]], 
         Cell[
          StyleData["SmallText", "Printout"], 
          CellMargins -> {{66, 10}, {7, 7}}, Hyphenation -> True]}, Closed]], 
      
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ResourcesText"], CellMargins -> {{66, 10}, {2, 0}}, 
          ReturnCreatesNewCell -> True, FontFamily -> "Source Sans Pro", 
          FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain", 
          FontColor -> GrayLevel[0]], 
         Cell[
          StyleData["ResourcesText", "Printout"], 
          CellMargins -> {{66, 10}, {2, 0}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ResourcesLink"], CellMargins -> {{66, 10}, {2, 0}}, 
          ReturnCreatesNewCell -> True, FontFamily -> "Source Sans Pro", 
          FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Italic", 
          FontColor -> GrayLevel[0]], 
         Cell[
          StyleData["ResourcesLink", "Printout"], 
          CellMargins -> {{66, 10}, {2, 0}}]}, Open]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Input"], CellMargins -> {{99, 10}, {3, 7}}, CellEpilog :> 
          Quiet[Needs["WolframBookTools`"]; 
            WolframBookTools`AddCaptionsToRegeneratedOutput[]], SpanMaxSize -> 
          1, FontSize -> 15], 
         Cell[
          StyleData["Input", "Printout"], CellMargins -> {{99, 10}, {3, 7}}]},
         Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Output"], CellMargins -> {{99, 10}, {14, 7}}, FontSize -> 
          15], 
         Cell[
          StyleData["Output", "Printout"], 
          CellMargins -> {{99, 10}, {14, 7}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["InputOnly"], CellMargins -> {{99, 10}, {25, 10}}, 
          SpanMaxSize -> 1, MenuSortingValue -> 1500, FontSize -> 15], 
         Cell[
          StyleData["InputOnly", "Printout"], 
          CellMargins -> {{99, 10}, {15, 10}}]}, Closed]], 
      Cell[
       StyleData["InitializationCell"], Background -> None], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["CellLabel"], 
          CellLabelMargins -> {{18, Inherited}, {Inherited, Inherited}}, 
          LanguageCategory -> None, StyleMenuListing -> None, FontFamily -> 
          "Roboto", FontSize -> 9, FontWeight -> "Regular", FontSlant -> 
          "Plain", FontColor -> GrayLevel[0.5]], 
         Cell[
          StyleData["CellLabel", "Printout"], FontSize -> 8, FontColor -> 
          GrayLevel[0.3]]}, Closed]], 
      Cell[
       StyleData[
       "CellLabelExpired", StyleDefinitions -> StyleData["CellLabel"]], 
       CellLabelMargins -> 6, FontSlant -> Italic, FontTracking -> 0.4, 
       FontColor -> GrayLevel[0.3]], 
      Cell[
       StyleData["IconizedDefaultName"], FontSize -> 12, 
       PrivateFontOptions -> {"OperatorSubstitution" -> False}, FontColor -> 
       GrayLevel[0.15]], 
      Cell[
       StyleData["IconizedFrame"], 
       TemplateBoxOptions -> {
        DisplayFunction -> (
          FrameBox[#, BaselinePosition -> Baseline, 
           FrameMargins -> {{0, 0}, {1, 1}}, FrameStyle -> GrayLevel[0.65], 
           Background -> GrayLevel[0.96], RoundingRadius -> 4, 
           DefaultBaseStyle -> {
            "DialogStyle", FontFamily -> "Roboto", FontSize -> 11}]& ), 
         Tooltip -> None}], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Message"], Editable -> False, 
          CellMargins -> {{66, Inherited}, {Inherited, Inherited}}, 
          CellGroupingRules -> "OutputGrouping", TextClipboardType -> 
          "PlainText", StripStyleOnPaste -> True, PageBreakWithin -> False, 
          GroupPageBreakWithin -> False, GeneratedCell -> True, 
          CellAutoOverwrite -> True, ShowCellLabel -> False, 
          DefaultFormatType -> DefaultOutputFormatType, ContextMenu -> 
          FEPrivate`FrontEndResource["ContextMenus", "Output"], 
          ShowAutoStyles -> False, ShowSyntaxStyles -> False, 
          AutoStyleOptions -> {
           "StringStyle" -> None, "SyntaxErrorStyle" -> None}, 
          HyphenationOptions -> {"HyphenationCharacter" -> "\[Continuation]"},
           AutoItalicWords -> {}, AutoQuoteCharacters -> {}, 
          PasteAutoQuoteCharacters -> {}, LanguageCategory -> None, 
          FormatType -> InputForm, ShowStringCharacters -> False, 
          CounterIncrements -> "Message", MenuSortingValue -> 1600, 
          StyleMenuListing -> None, FontFamily :> 
          CurrentValue["PanelFontFamily"], FontSize -> 11, FontColor -> 
          RGBColor[0.5072, 0.16933, 0.15221]], 
         Cell[
          StyleData["Message", "Printout"], 
          CellMargins -> {{100, Inherited}, {Inherited, Inherited}}, 
          CellFrameLabels -> {{None, None}, {None, None}}, FontColor -> 
          GrayLevel[0]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ExerciseInput", StyleDefinitions -> StyleData["Input"]], 
          CellMargins -> {{106, 10}, {4, 7}}, 
          GeneratedCellStyles -> {"Output" -> "ExerciseOutput"}, 
          ShowCellLabel -> False, SpanMaxSize -> 1, FontSize -> 14], 
         Cell[
          StyleData["ExerciseInput", "Printout"], 
          CellMargins -> {{106, 10}, {4, 7}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "ExerciseOutput", StyleDefinitions -> StyleData["Output"]], 
          CellMargins -> {{106, 10}, {10, 4}}, ShowCellLabel -> False, 
          FontSize -> 14], 
         Cell[
          StyleData["ExerciseOutput", "Printout"], 
          CellMargins -> {{106, 10}, {10, 4}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Picture"], CellMargins -> {{66, 10}, {10, 10}}, 
          CellGroupingRules -> "GraphicsGrouping", StyleMenuListing -> None], 
         
         Cell[
          StyleData["Picture", "Printout"], 
          CellMargins -> {{66, 10}, {8, 8}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Comment"], FontFamily -> "Verdana", FontSize -> 13, 
          FontColor -> RGBColor[0.3, 0.3, 0.2], Background -> 
          RGBColor[1, 0.9, 0.8]], 
         Cell[
          StyleData["Comment", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Suggestion"], 
          CellFrameLabels -> {{Inherited, Inherited}, {Inherited, 
             Cell[
              BoxData[
               TagBox[
                GridBox[{{
                   ButtonBox[
                   "\"  Insert  \"", 
                    ButtonFunction :> (Needs["DocuTools`"]; 
                    DocuTools`DocuToolsLoader[]; 
                    DocuTools`ReplaceWithSuggestion[]), Evaluator -> 
                    Automatic, 
                    BaseStyle -> {FontSize -> 10, FontFamily -> "Verdana"}, 
                    ContentPadding -> False, Appearance -> Automatic, Method -> 
                    "Queued"], 
                   ButtonBox[
                   "\"  Stet  \"", 
                    ButtonFunction :> (Needs["DocuTools`"]; 
                    DocuTools`DocuToolsLoader[]; DocuTools`StetSuggestion[]), 
                    Evaluator -> Automatic, 
                    BaseStyle -> {FontSize -> 10, FontFamily -> "Verdana"}, 
                    Appearance -> Automatic, Method -> "Queued"], "    ", 
                   ButtonBox[
                   "\"  Next  \"", 
                    ButtonFunction :> (Needs["DocuTools`"]; 
                    DocuTools`DocuToolsLoader[]; 
                    DocuTools`FindNextSuggestion[]), Evaluator -> Automatic, 
                    BaseStyle -> {FontSize -> 10, FontFamily -> "Verdana"}, 
                    Appearance -> Automatic, Method -> "Queued"], 
                   ButtonBox[
                   "\"  Previous  \"", 
                    ButtonFunction :> (Needs["DocuTools`"]; 
                    DocuTools`DocuToolsLoader[]; 
                    DocuTools`FindPreviousSuggestion[]), Evaluator -> 
                    Automatic, 
                    BaseStyle -> {FontSize -> 10, FontFamily -> "Verdana"}, 
                    Appearance -> Automatic, Method -> "Queued"], "\"\""}}, 
                 AutoDelete -> False, 
                 GridBoxItemSize -> {
                  "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], 
                "Grid"]], ShowStringCharacters -> False, Background -> 
              None]}}, Background -> RGBColor[1, 0.8, 0.8]], 
         Cell[
          StyleData["Suggestion", "Printout"], 
          CellElementSpacings -> {"CellMinHeight" -> 0.1}, CellOpen -> False, 
          CellFrameMargins -> 0, FontSize -> 0.1, Magnification -> 0.01]}, 
        Open]], 
      Cell[
       StyleData["Tentative"], 
       CellBracketOptions -> {
        "Color" -> RGBColor[
          0.30000000000000016`, 0.6000000000000003, 0.7000000000000003], 
         "Thickness" -> 5}, Evaluatable -> False, CellFrameMargins -> 4, 
       StyleMenuListing -> None, Background -> 
       RGBColor[0.8500000000000003, 0.9500000000000003, 1]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["PrintOnly"], 
          CellBracketOptions -> {
           "Color" -> RGBColor[0.89, 0.867, 0.99], "Thickness" -> 5}, 
          CellFrameMargins -> 4, StyleMenuListing -> None, Background -> 
          RGBColor[0.89, 0.867, 0.99]], 
         Cell[
          StyleData["PrintOnly", "Printout"], 
          CellElementSpacings -> {"CellMinHeight" -> 0.1}, CellOpen -> False, 
          CellFrameMargins -> 0, FontSize -> 0.1, Magnification -> 0.01], 
         Cell[
          StyleData["Tentative", "Printout"], 
          CellElementSpacings -> {"CellMinHeight" -> 0.1}, CellOpen -> False, 
          CellFrameMargins -> 0, FontSize -> 0.1, Magnification -> 0.01]}, 
        Open]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["WebOnly"], 
          CellBracketOptions -> {
           "Color" -> RGBColor[0.91, 0.92, 0.9], "Thickness" -> 5}, 
          CellFrameMargins -> 4, StyleMenuListing -> None, Background -> 
          RGBColor[0.91, 0.92, 0.86]], 
         Cell[
          StyleData["WebOnly", "Printout"], 
          CellElementSpacings -> {"CellMinHeight" -> 0.1}, CellOpen -> False, 
          CellFrameMargins -> 0, FontSize -> 0.1, Magnification -> 0.01]}, 
        Open]], 
      Cell[
       StyleData["Lockdown"], 
       CellBracketOptions -> {
        "Color" -> RGBColor[0.8, 0.5, 0.4], "Thickness" -> 3}, Evaluatable -> 
       False, StyleMenuListing -> None], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["IndexField"], CellFrame -> True, CellFrameColor -> 
          GrayLevel[0.7], FontFamily -> "Verdana", FontSize -> 12], 
         Cell[
          StyleData["IndexField", "Invisibles"], CellFrame -> False, 
          ShowCellBracket -> False, CellOpen -> False, FontSize -> 1]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ExplanationField"], CellFrame -> True, CellFrameColor -> 
          GrayLevel[0.7], FontFamily -> "Verdana", FontSize -> 12], 
         Cell[
          StyleData["ExplanationField", "Invisibles"], CellFrame -> False, 
          ShowCellBracket -> False, CellOpen -> False, FontSize -> 1]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["EnglishHighlight"], FontFamily -> "WolframCodeSans"], 
         Cell[
          StyleData["EnglishHighlight", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SolvedExample"], CellMargins -> {{66, 30}, {0, 16}}, 
          CellBracketOptions -> {"OverlapContent" -> True}, 
          CellFrameMargins -> {{25, 13}, {10, 25}}, 
          LightDarkAutoColorRules -> <|"AllColors" -> True|>, 
          CounterIncrements -> "SolvedExample", FontFamily -> 
          "Source Sans Pro", FontSize -> 16.5, FontWeight -> "SemiBold", 
          FontColor -> 
          RGBColor[
           0.050002574846150734`, 0.050002574846150734`, 
            0.050002574846150734`], Background -> Hue[0.085, 0.07, 0.95]], 
         Cell[
          StyleData["SolvedExample", "Printout"], 
          CellMargins -> {{66, 0}, {0, 16}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "SolvedExampleNote", StyleDefinitions -> StyleData["Text"]], 
          CellMargins -> {{66, 30}, {0, 0}}, 
          CellBracketOptions -> {"OverlapContent" -> True}, 
          CellFrameMargins -> {{25, 13}, {10, 10}}, 
          LightDarkAutoColorRules -> <|"AllColors" -> True|>, FontSize -> 17, 
          FontColor -> 
          RGBColor[
           0.050002574846150734`, 0.050002574846150734`, 
            0.050002574846150734`], Background -> Hue[0.085, 0.07, 0.95]], 
         Cell[
          StyleData["SolvedExampleNote", "Printout"], 
          CellMargins -> {{66, 0}, {0, -6}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "SolvedExampleDisplayFormula", StyleDefinitions -> 
           StyleData["DisplayFormula"]], CellMargins -> {{66, 30}, {0, 0}}, 
          CellBracketOptions -> {"OverlapContent" -> True}, 
          CellFrameMargins -> {{32, Inherited}, {12, 8}}, TextAlignment -> 
          Left, LightDarkAutoColorRules -> <|"AllColors" -> True|>, FontSize -> 
          14, FontColor -> 
          RGBColor[
           0.050002574846150734`, 0.050002574846150734`, 
            0.050002574846150734`], Background -> Hue[0.085, 0.07, 0.95]], 
         Cell[
          StyleData["SolvedExampleDisplayFormula", "Printout"], 
          CellMargins -> {{66, 0}, {0, -6}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "SolvedExampleDisplayFormulaNumbered", StyleDefinitions -> 
           StyleData["DisplayFormula"]], CellMargins -> {{66, 40}, {0, 0}}, 
          CellBracketOptions -> {"OverlapContent" -> True}, 
          CellFrameMargins -> {{32, 46}, {12, 8}}, 
          CellFrameLabels -> {{None, 
             Cell[
              TextData[{"(", 
                CounterBox["SolvedExampleDisplayFormulaNumbered"], ")"}], 
              "DisplayFormulaEquationNumber", CellSize -> {36, Automatic}]}, {
            None, None}}, 
          CellFrameLabelMargins -> {{Inherited, -46}, {Inherited, Inherited}},
           TextAlignment -> Left, 
          LightDarkAutoColorRules -> <|"AllColors" -> True|>, 
          CounterIncrements -> "SolvedExampleDisplayFormulaNumbered", 
          FontSize -> 14, FontColor -> 
          RGBColor[
           0.050002574846150734`, 0.050002574846150734`, 
            0.050002574846150734`], Background -> Hue[0.085, 0.07, 0.95]], 
         Cell[
          StyleData["SolvedExampleDisplayFormulaNumbered", "Printout"], 
          CellMargins -> {{66, 0}, {0, -6}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "SolvedExampleInput", StyleDefinitions -> StyleData["Input"]], 
          CellMargins -> {{66, 30}, {0, 0}}, 
          CellBracketOptions -> {"OverlapContent" -> True}, 
          GeneratedCellStyles -> {
           "Message" -> "SolvedExampleMessage", "Output" -> 
            "SolvedExampleOutput"}, 
          CellFrameMargins -> {{32, Inherited}, {3, 7}}, 
          LightDarkAutoColorRules -> <|"AllColors" -> True|>, FontColor -> 
          RGBColor[
           0.050002574846150734`, 0.050002574846150734`, 
            0.050002574846150734`], Background -> Hue[0.085, 0.07, 0.95]], 
         Cell[
          StyleData["SolvedExampleInput", "Printout"], 
          CellMargins -> {{66, 0}, {0, -7}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "SolvedExampleOutput", StyleDefinitions -> StyleData["Output"]], 
          CellMargins -> {{66, 30}, {0, 0}}, 
          CellBracketOptions -> {"OverlapContent" -> True}, 
          CellFrameMargins -> {{32, Inherited}, {14, 7}}, 
          LightDarkAutoColorRules -> <|"AllColors" -> True|>, FontColor -> 
          RGBColor[
           0.050002574846150734`, 0.050002574846150734`, 
            0.050002574846150734`], Background -> Hue[0.085, 0.07, 0.95]], 
         Cell[
          StyleData["SolvedExampleOutput", "Printout"], 
          CellMargins -> {{66, 0}, {6, -6}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["SolvedExampleMessage"], 
          CellMargins -> {{66, 30}, {0, 0}}, 
          CellBracketOptions -> {"OverlapContent" -> True}, 
          CellFrameMargins -> {{25, 13}, {10, 10}}, 
          LightDarkAutoColorRules -> <|"AllColors" -> True|>, FontColor -> 
          RGBColor[
           0.050002574846150734`, 0.050002574846150734`, 
            0.050002574846150734`], Background -> Hue[0.085, 0.07, 0.95]], 
         Cell[
          StyleData["SolvedExampleMessage", "Printout"], 
          CellMargins -> {{66, 0}, {0, -6}}]}, Closed]], 
      Cell[
       StyleData["SolvedExampleEndCap"], CellMargins -> {{66, 30}, {0, 0}}, 
       CellBracketOptions -> {"OverlapContent" -> True}, 
       CellElementSpacings -> {"CellMinHeight" -> 20}, 
       CellFrameMargins -> {{10, 13}, {0, 0}}, 
       LightDarkAutoColorRules -> <|"AllColors" -> True|>, FontSize -> 2, 
       FontColor -> 
       RGBColor[
        0.050002574846150734`, 0.050002574846150734`, 0.050002574846150734`], 
       Background -> Hue[0.085, 0.07, 0.95]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Theorem"], Editable -> False, 
          CellMargins -> {{66, 30}, {0, 16}}, 
          CellBracketOptions -> {"OverlapContent" -> True}, 
          CellFrameMargins -> {{25, 13}, {0, 25}}, 
          CellChangeTimes -> {3.968922600990673*^9}, 
          LightDarkAutoColorRules -> <|"AllColors" -> True|>, 
          CounterIncrements -> "Subsection", FontFamily -> "Source Sans Pro", 
          FontSize -> 16.5, FontWeight -> "SemiBold", FontColor -> 
          RGBColor[
           0.00010000076130382223`, 0.007800042619928998, 
            0.13000297516585294`], Background -> Hue[0.57, 0.098, 0.996], 
          GridBoxOptions -> {
           GridBoxAlignment -> {
             "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, 
            GridBoxSpacings -> {
             "Columns" -> {{Automatic}}, "RowsIndexed" -> {2 -> 0}}}, 
          GraphicsBoxOptions -> {
           ImagePadding -> None, ImageSize -> {900, 1}, PlotRangePadding -> 
            0}], 
         Cell[
          StyleData["Theorem", "Printout"], CellMargins -> {{66, 0}, {0, 16}},
           CellChangeTimes -> {{3.965900023638689*^9, 
           3.965900023829352*^9}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "TheoremStatement", StyleDefinitions -> StyleData["Text"]], 
          CellMargins -> {{66, 30}, {0, 0}}, 
          CellBracketOptions -> {"OverlapContent" -> True}, 
          CellFrameMargins -> {{25, 13}, {10, 10}}, 
          LightDarkAutoColorRules -> <|"AllColors" -> True|>, FontSize -> 17, 
          FontColor -> 
          RGBColor[
           0.00010000076130382223`, 0.007800042619928998, 
            0.13000297516585294`], Background -> Hue[0.57, 0.098, 0.996]], 
         Cell[
          StyleData["TheoremStatement", "Printout"], 
          CellMargins -> {{66, 0}, {0, 0}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "ProofTheoremDisplayFormula", StyleDefinitions -> 
           StyleData["DisplayFormula"]], CellMargins -> {{66, 30}, {0, 0}}, 
          CellBracketOptions -> {"OverlapContent" -> True}, 
          CellFrameMargins -> {{32, Inherited}, {12, 8}}, 
          CellFrameLabelMargins -> {{Inherited, -26}, {Inherited, Inherited}},
           TextAlignment -> Left, 
          LightDarkAutoColorRules -> <|"AllColors" -> True|>, 
          MenuSortingValue -> 1710, FontFamily -> "Baskerville", FontSize -> 
          14, FontWeight -> "Regular", FontSlant -> "Plain", 
          PrivateFontOptions -> {"FontPostScriptName" -> "Automatic"}, 
          FontColor -> 
          RGBColor[
           0.00010000076130382223`, 0.007800042619928998, 
            0.13000297516585294`], Background -> Hue[0.57, 0.098, 0.996]], 
         Cell[
          StyleData["ProofTheoremDisplayFormula", "Printout"], 
          CellMargins -> {{66, 0}, {0, 0}}]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "ProofTheoremDisplayFormulaNumbered", StyleDefinitions -> 
           StyleData["DisplayFormula"]], CellMargins -> {{66, 40}, {0, 0}}, 
          CellBracketOptions -> {"OverlapContent" -> True}, 
          CellFrameMargins -> {{32, 46}, {12, 8}}, 
          CellFrameLabels -> {{None, 
             Cell[
              TextData[{"(", 
                CounterBox["ProofTheoremDisplayFormulaNumbered"], ")"}], 
              "DisplayFormulaEquationNumber", CellSize -> {36, Automatic}]}, {
            None, None}}, 
          CellFrameLabelMargins -> {{Inherited, -46}, {Inherited, Inherited}},
           TextAlignment -> Left, 
          LightDarkAutoColorRules -> <|"AllColors" -> True|>, 
          CounterIncrements -> "ProofTheoremDisplayFormulaNumbered", 
          MenuSortingValue -> 1710, FontFamily -> "Baskerville", FontSize -> 
          14, FontWeight -> "Regular", FontSlant -> "Plain", 
          PrivateFontOptions -> {"FontPostScriptName" -> "Automatic"}, 
          FontColor -> 
          RGBColor[
           0.00010000076130382223`, 0.007800042619928998, 
            0.13000297516585294`], Background -> Hue[0.57, 0.098, 0.996]], 
         Cell[
          StyleData["ProofTheoremDisplayFormulaNumbered", "Printout"], 
          CellMargins -> {{66, 0}, {0, 0}}]}, Open]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Proof"], Editable -> False, 
          CellMargins -> {{66, 30}, {0, 0}}, 
          CellBracketOptions -> {"OverlapContent" -> True}, 
          CellFrameMargins -> {{25, 13}, {0, 20}}, 
          LightDarkAutoColorRules -> <|"AllColors" -> True|>, FontFamily -> 
          "Source Sans Pro", FontSize -> 16.5, FontWeight -> "SemiBold", 
          FontColor -> 
          RGBColor[
           0.00010000076130382223`, 0.007800042619928998, 
            0.13000297516585294`], Background -> Hue[0.57, 0.098, 0.996], 
          GridBoxOptions -> {
           GridBoxAlignment -> {
             "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, 
            GridBoxSpacings -> {
             "Columns" -> {{Automatic}}, "RowsIndexed" -> {2 -> 0}}}, 
          GraphicsBoxOptions -> {
           ImagePadding -> None, ImageSize -> {900, 1}, PlotRangePadding -> 
            0}], 
         Cell[
          StyleData["Proof", "Printout"], CellMargins -> {{66, 0}, {0, 0}}]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["ProofContent", StyleDefinitions -> StyleData["Text"]], 
          CellMargins -> {{66, 30}, {0, 0}}, 
          CellBracketOptions -> {"OverlapContent" -> True}, 
          CellFrameMargins -> {{25, 13}, {10, 10}}, 
          LightDarkAutoColorRules -> <|"AllColors" -> True|>, FontSize -> 17, 
          FontColor -> 
          RGBColor[
           0.00010000076130382223`, 0.007800042619928998, 
            0.13000297516585294`], Background -> Hue[0.57, 0.098, 0.996]], 
         Cell[
          StyleData["ProofContent", "Printout"], 
          CellMargins -> {{66, 0}, {0, 0}}]}, Closed]], 
      Cell[
       StyleData["ProofTheoremEndCap"], CellMargins -> {{66, 30}, {0, 0}}, 
       CellBracketOptions -> {"OverlapContent" -> True}, 
       CellElementSpacings -> {"CellMinHeight" -> 20}, 
       CellFrameMargins -> {{10, 13}, {0, 0}}, FontSize -> 2, FontColor -> 
       RGBColor[
        0.00010000076130382223`, 0.007800042619928998, 0.13000297516585294`], 
       Background -> Hue[0.57, 0.098, 0.996]], 
      Cell[
       StyleData["Banner", StyleDefinitions -> StyleData["Text"]], Editable -> 
       False, WindowToolbars -> {}, PageWidth :> WindowWidth, 
       Alignment -> {{Left, Right}, {Center, Center}}, 
       CellFrame -> {{0, 1}, {0, 0}}, ShowCellBracket -> False, 
       CellMargins -> {{0, 0}, {0, 0}}, 
       CellBracketOptions -> {"OverlapContent" -> True}, 
       CellGroupingRules -> {"SectionGrouping", 0}, CellContext -> CellGroup, 
       CellFrameMargins -> {{0, 20}, {5, 0}}, CellFrameColor -> 
       RGBColor[
        0.00009999889888604668, 0.1607913972623736, 0.4000129703059972], 
       ShowStringCharacters -> False, FontFamily -> "Source Sans Pro", 
       FontSize -> 20, FontWeight -> "SemiBold", FontColor -> GrayLevel[1], 
       Background -> 
       RGBColor[
        0.00009999889888604668, 0.1607913972623736, 0.4000129703059972]], 
      Cell[
       StyleData[
       "BannerCitationOpenerLabel", StyleDefinitions -> StyleData["Text"]], 
       WindowToolbars -> {}, Selectable -> True, PageWidth :> WindowWidth, 
       CellFrame -> {{0, 1}, {0, 0}}, ShowCellBracket -> False, 
       CellMargins -> {{0, 0}, {0, 0}}, 
       CellGroupingRules -> {"SectionGrouping", 70}, CellContext -> CellGroup,
        CellFrameMargins -> {{28, 0}, {4, 5}}, CellFrameColor -> 
       RGBColor[0.8627719122171761, 0.9333285528421761, 0.9999789434671761], 
       ShowStringCharacters -> False, FontFamily -> "Source Sans Pro", 
       FontSize -> 11, FontWeight -> "Regular", FontColor -> 
       RGBColor[
        0.27843797333540327`, 0.40392625458540327`, 0.5137851263583065], 
       Background -> 
       RGBColor[0.8627719122171761, 0.9333285528421761, 0.9999789434671761]], 
      
      Cell[
       StyleData["BannerCitation", StyleDefinitions -> StyleData["Text"]], 
       WindowToolbars -> {}, Selectable -> True, PageWidth :> WindowWidth, 
       CellFrame -> {{0, 1}, {0, 0}}, ShowCellBracket -> False, 
       CellMargins -> {{0, 0}, {0, -1}}, 
       CellElementSpacings -> {"CellMinHeight" -> 2, "ClosedCellHeight" -> 1},
        Copyable -> True, CellContext -> CellGroup, 
       CellFrameMargins -> {{28, 0}, {8, 0}}, CellFrameColor -> 
       RGBColor[0.8627719122171761, 0.9333285528421761, 0.9999789434671761], 
       ShowStringCharacters -> False, FontFamily -> "Source Sans Pro", 
       FontSize -> 11, FontWeight -> "Bold", FontColor -> 
       RGBColor[
        0.27843797333540327`, 0.40392625458540327`, 0.5137851263583065], 
       Background -> 
       RGBColor[0.8627719122171761, 0.9333285528421761, 0.9999789434671761]], 
      
      Cell[
       StyleData[
       "BannerPreReleaseLabel", StyleDefinitions -> 
        StyleData["BannerCitationOpenerLabel"]], 
       CellFrameMargins -> {{0, 30}, {2, 0}}, 
       ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{
            
            NotebookLocate[#2, "OpenInNewWindow" -> 
             CurrentValue["HyperlinkModifierKey"]]}]& ), Evaluator -> None, 
         Method -> "Queued"}], 
      Cell[
       StyleData["BannerHeading"], MenuSortingValue -> None, FontFamily -> 
       "Source Sans Pro", FontSize -> 11, FontWeight -> "Regular", FontColor -> 
       RGBColor[
        0.9999903558940054, 0.5650486394018294, 0.14501301402226988`]]}, 
     Open]], 
   Cell[
    CellGroupData[{
      Cell[
      "Inline and special formatting", "Section", 
       CellChangeTimes -> {{3.646681732592888*^9, 3.646681749414035*^9}}], 
      Cell[
       StyleData["DateObject"], ContextMenu -> 
       FEPrivate`FrontEndResource["ContextMenus", "DateObject"], 
       TemplateBoxOptions -> {DisplayFunction -> (FrameBox[
           
           StyleBox[#, "Text", FontFamily -> "Source Sans Pro", FontSize -> 
            13, FontWeight -> "Plain", FontColor -> 
            RGBColor[
             0.12941176470588237`, 0.34509803921568627`, 
              0.43529411764705883`], ShowStringCharacters -> False], 
           Background -> Hue[0.11, 0, 0.97], FrameMargins -> {{5, 5}, {2, 2}},
            FrameStyle -> GrayLevel[0.65], RoundingRadius -> 2, 
           BaselinePosition -> Baseline, 
           BaseStyle -> {FontFamily -> "Courier"}]& ), 
         InterpretationFunction -> (#2& )}], 
      Cell[
       StyleData["DateObjectLabel"], FontWeight -> "Plain", FontColor -> 
       GrayLevel[0.2]], 
      Cell[
       StyleData["SummaryItemAnnotation"], FontColor -> GrayLevel[0.3]], 
      Cell[
       StyleData["Caption"], FontFamily :> CurrentValue["PanelFontFamily"], 
       FontSize :> 0.8 Inherited, FontWeight -> "Plain", FontSlant -> "Plain",
        FontColor :> CurrentValue[{"SystemColors", "DialogText"}], 
       TemplateBoxOptions -> {DisplayFunction -> (ButtonBox[
           
           FrameBox[#, FrameStyle -> GrayLevel[0.5], Background -> 
            GrayLevel[0.969], FrameMargins -> 7, RoundingRadius -> 3, 
            BaselinePosition -> Baseline], Active -> False, Alignment -> Left,
            Appearance -> None, BaseStyle -> {Deployed -> False}, 
           DefaultBaseStyle -> {"Pane", Background -> None}, BaselinePosition -> 
           Baseline]& ), Tooltip -> None}, 
       GraphicsBoxOptions -> {
        Axes -> False, 
         DefaultBaseStyle -> {
          "GraphicsNoFontsOrMenu", CacheGraphics -> False}, ImageSize -> 
         Dynamic[{
           Automatic, 
            3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
             Magnification])}]}, 
       Graphics3DBoxOptions -> {
        Axes -> False, DefaultBaseStyle -> {"Graphics3DNoFontsOrMenu"}, 
         ImageSize -> 
         Dynamic[{
           Automatic, 
            3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
             Magnification])}]}], 
      Cell[
       StyleData["GraphicsEmbeddedCaption"], LineIndent -> 0, FontFamily -> 
       "Source Sans Pro", FontSize -> 14], 
      Cell[
       StyleData["SummaryPanel"], FontFamily :> 
       CurrentValue["PanelFontFamily"], FontSize :> 0.8 Inherited, FontWeight -> 
       "Plain", FontSlant -> "Plain", FontColor :> 
       CurrentValue[{"SystemColors", "DialogText"}], 
       TemplateBoxOptions -> {DisplayFunction -> (ButtonBox[
           
           FrameBox[#, FrameStyle -> GrayLevel[0.5], Background -> 
            GrayLevel[0.969], FrameMargins -> 7, RoundingRadius -> 3, 
            BaselinePosition -> Baseline], Active -> False, Alignment -> Left,
            Appearance -> None, BaseStyle -> {Deployed -> False}, 
           DefaultBaseStyle -> {"Pane", Background -> None}, BaselinePosition -> 
           Baseline]& ), Tooltip -> None}, 
       GraphicsBoxOptions -> {
        Axes -> False, 
         DefaultBaseStyle -> {
          "GraphicsNoFontsOrMenu", CacheGraphics -> False}, ImageSize -> 
         Dynamic[{
           Automatic, 
            3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
             Magnification])}]}, 
       Graphics3DBoxOptions -> {
        Axes -> False, DefaultBaseStyle -> {"Graphics3DNoFontsOrMenu"}, 
         ImageSize -> 
         Dynamic[{
           Automatic, 
            3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
             Magnification])}]}], 
      Cell[
       StyleData["Graphics"], MenuSortingValue -> None], 
      Cell[
       StyleData["Graphics3D"], MenuSortingValue -> None], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["StandardForm"], LineSpacing -> {1.25, 0, 2}, FontFamily -> 
          "Source Sans Code"], 
         Cell[
          StyleData["StandardForm", "Printout"], LineSpacing -> {1.25, 0, 2}, 
          FontFamily -> "Source Sans Code"]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["OutputForm"], LineSpacing -> {0.6, 1}, FontFamily -> 
          "Source Sans Code"], 
         Cell[
          StyleData["OutputForm", "Printout"], LineSpacing -> {0.6, 1}, 
          FontFamily -> "Source Sans Code"]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["InputForm"], FontFamily -> "Source Sans Code"], 
         Cell[
          StyleData["InputForm", "Printout"], FontFamily -> 
          "Source Sans Code"]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["FormattedFunction"], FontFamily -> "Source Sans Pro", 
          FontSize :> If[CurrentValue[
              EvaluationCell[], "FontFamily"] === "Source Sans Pro", 
            Inherited, 
            If[CurrentValue[
               EvaluationCell[], "FontFamily"] === "WolframCodeSans", 
             Inherited, Inherited 0.9]]], 
         Cell[
          StyleData["FormattedFunction", "Printout"], FontFamily -> 
          "Source Sans Pro", FontSize :> If[CurrentValue[
              EvaluationCell[], "FontFamily"] === "Source Sans Pro", 
            Inherited, 
            If[CurrentValue[
               EvaluationCell[], "FontFamily"] === "WolframCodeSans", 
             Inherited, Inherited 0.88]]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Link"], FontFamily -> "Source Sans Pro", FontSize :> 
          If[CurrentValue[
              EvaluationCell[], "FontFamily"] === "Source Sans Pro", 
            Inherited, 
            If[CurrentValue[
               EvaluationCell[], "FontFamily"] === "WolframCodeSans", 
             Inherited, Inherited 0.9]], FontColor -> GrayLevel[0], 
          ButtonBoxOptions -> {
           Active -> True, Appearance -> {Automatic, None}, 
            ButtonFunction :> (FEPrivate`FrontEndExecute[
              FrontEndToken["OpenHelpLink", {#}]]& ), ButtonNote -> 
            ButtonData, Evaluator -> None, ImageSize -> Full, Method -> 
            "Preemptive"}], 
         Cell[
          StyleData["Link", "Printout"], FontFamily -> "Source Sans Pro", 
          FontSize :> If[CurrentValue[
              EvaluationCell[], "FontFamily"] === "Source Sans Pro", 
            Inherited, 
            If[CurrentValue[
               EvaluationCell[], "FontFamily"] === "WolframCodeSans", 
             Inherited, Inherited 0.88]], FontColor -> GrayLevel[0], 
          ButtonBoxOptions -> {
           Active -> True, Appearance -> {Automatic, None}, 
            ButtonFunction :> (FEPrivate`FrontEndExecute[
              FrontEndToken["OpenHelpLink", {#}]]& ), ButtonNote -> 
            ButtonData, Evaluator -> None, ImageSize -> Full, Method -> 
            "Preemptive"}]}, Open]], 
      Cell[
       StyleData[
       "CaptionFunctionLink", StyleDefinitions -> StyleData["Link"]], 
       FontFamily -> "Source Sans Pro", FontSize -> 14, FontColor -> 
       GrayLevel[0]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["DetailsLink"], FontFamily -> "Source Sans Pro", FontSize :> 
          If[CurrentValue[
              EvaluationCell[], "FontFamily"] === "Source Sans Pro", 
            Inherited, 
            If[CurrentValue[
               EvaluationCell[], "FontFamily"] === "WolframCodeSans", 
             Inherited, Inherited 0.9]], FontColor -> GrayLevel[0], 
          ButtonBoxOptions -> {
           ButtonFunction :> (
             Function[$CellContext`t, 
              NotebookFind[$CellContext`t, "Notes", All, CellStyle]; 
              NotebookFind[$CellContext`t, "NotesSection", All, CellStyle, 
                AutoScroll -> Top]][
              Documentation`HelpLookup["ref/" <> #]]& ), ButtonSource -> 
            ButtonContents, Evaluator -> Automatic}], 
         Cell[
          StyleData["DetailsLink", "Printout"], FontFamily -> 
          "Source Sans Pro", FontSize :> If[CurrentValue[
              EvaluationCell[], "FontFamily"] === "Source Sans Pro", 
            Inherited, 
            If[CurrentValue[
               EvaluationCell[], "FontFamily"] === "WolframCodeSans", 
             Inherited, Inherited 0.88]], FontColor -> GrayLevel[0]]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["LinkAlt"], FontFamily -> "Source Sans Pro", FontSize :> 
          If[CurrentValue[
              EvaluationCell[], "FontFamily"] === "Source Sans Pro", 
            Inherited, 
            If[CurrentValue[
               EvaluationCell[], "FontFamily"] === "WolframCodeSans", 
             Inherited, Inherited 0.9]], FontColor -> GrayLevel[0]], 
         Cell[
          StyleData["LinkAlt", "Printout"], FontFamily -> "Source Sans Pro", 
          FontSize :> If[CurrentValue[
              EvaluationCell[], "FontFamily"] === "Source Sans Pro", 
            Inherited, 
            If[CurrentValue[
               EvaluationCell[], "FontFamily"] === "WolframCodeSans", 
             Inherited, Inherited 0.88]], FontColor -> GrayLevel[0]]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Hyperlink"], MenuSortingValue -> None, FontColor -> 
          RGBColor[0.3, 0.35, 0.5]], 
         Cell[
          StyleData["Hyperlink", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["FunctionLink", StyleDefinitions -> StyleData["Link"]], 
          FontColor -> GrayLevel[0]], 
         Cell[
          StyleData["FunctionLink", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "FunctionHyperlink", StyleDefinitions -> StyleData["Link"]], 
          FontColor -> GrayLevel[0], 
          ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{
               
               NotebookLocate[#2, "OpenInNewWindow" -> 
                CurrentValue["HyperlinkModifierKey"]]}]& ), Evaluator -> None,
             Method -> "Queued"}], 
         Cell[
          StyleData["FunctionHyperlink", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData[
          "ResourceFunctionlink", StyleDefinitions -> StyleData["Hyperlink"]],
           MenuSortingValue -> None, FontFamily -> "Source Sans Pro", 
          FontSize :> If[CurrentValue[
              EvaluationCell[], "FontFamily"] === "Source Sans Pro", 
            Inherited, 
            If[CurrentValue[
               EvaluationCell[], "FontFamily"] === "WolframCodeSans", 
             Inherited, Inherited 0.9]], FontColor -> GrayLevel[0]], 
         Cell[
          StyleData["ResourceFunctionlink", "Printout"], FontFamily -> 
          "Source Sans Pro", FontSize :> If[CurrentValue[
              EvaluationCell[], "FontFamily"] === "Source Sans Pro", 
            Inherited, 
            If[CurrentValue[
               EvaluationCell[], "FontFamily"] === "WolframCodeSans", 
             Inherited, Inherited 0.88]], FontColor -> GrayLevel[0]]}, 
        Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["Crosslink"], 
          ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{
               
               NotebookLocate[#2, "OpenInNewWindow" -> 
                CurrentValue["HyperlinkModifierKey"]]}]& ), Evaluator -> None,
             Method -> "Queued"}], 
         Cell[
          StyleData["Crosslink", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["CodeLink"], 
          ButtonBoxOptions -> {
           Active -> True, Appearance -> {Automatic, None}, 
            ButtonFunction :> (FEPrivate`FrontEndExecute[
              FrontEndToken["OpenHelpLink", {
                Part[#, 1]}]]& ), ButtonNote -> ButtonData, Evaluator -> None,
             ImageSize -> Full, Method -> "Preemptive"}], 
         Cell[
          StyleData["CodeLink", "Printout"]]}, Open]], 
      Cell[
       StyleData["VocabularyLink"], FontColor -> GrayLevel[0], 
       ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{
            
            NotebookLocate[#2, "OpenInNewWindow" -> 
             CurrentValue["HyperlinkModifierKey"]]}]& ), Evaluator -> None, 
         Method -> "Queued"}], 
      Cell[
       StyleData["InlineMath"], SpanMaxSize -> 0, CellID -> 644119144], 
      Cell[
       StyleData["InlineCode"], AutoSpacing -> True, SpanMaxSize -> 0, 
       FontColor -> GrayLevel[0]], 
      Cell[
       StyleData["InlineCodeText"], SpanMaxSize -> 0, FontFamily -> 
       "Source Sans Pro", FontSize -> 0.94 Inherited], 
      Cell[
       StyleData["InlineFormula"]], 
      Cell[
       StyleData["NoLabel"], 
       GeneratedCellStyles -> {"Output" -> "ExerciseOutput"}, ShowCellLabel -> 
       False], 
      Cell[
       StyleData["MenuName"], FontFamily -> "Verdana", FontWeight -> "Bold", 
       FontColor -> GrayLevel[0.286275]], 
      Cell[
       StyleData["DialogElementName"], 
       CellChangeTimes -> {3.642251807158844*^9, 3.8358753610849867`*^9, 
        3.835875471581566*^9}, FontFamily -> "Source Sans Pro SemiBold", 
       FontSize -> 0.95 Inherited, FontWeight -> "Regular"], 
      Cell[
       StyleData["KeyEvent"], CellFrame -> {{0.5, 0.5}, {0.5, 0.5}}, 
       CellFrameMargins -> {{2, 2}, {2, 2}}, CellFrameColor -> GrayLevel[0.7],
        FontFamily -> "Verdana", FontColor -> GrayLevel[0.4]], 
      Cell[
       StyleData["KeyEventDelimiter"], FontColor -> GrayLevel[0.5]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["CodeVariable"], FontFamily -> "Times", FontSize -> 
          1.05 Inherited, FontWeight -> "Plain", FontSlant -> "Italic"], 
         Cell[
          StyleData["CodeVariable", "Printout"]]}, Closed]], 
      Cell[
       CellGroupData[{
         Cell[
          StyleData["CodeVariableLrg"], FontFamily -> "Times", FontSize -> 
          1.1 Inherited, FontWeight -> "Plain", FontSlant -> "Italic"], 
         Cell[
          StyleData["CodeVariableLrg", "Printout"]]}, Open]], 
      Cell[
       CellGroupData[{
         Cell["Key Styles", "Subsubsection"], 
         Cell[
          StyleData["Key0"], 
          TemplateBoxOptions -> {DisplayFunction -> (FrameBox[
              TagBox[
               GridBox[{{
                  ItemBox[
                   
                   TemplateBox[{"\[ThickSpace]", #, "\[ThickSpace]"}, 
                    "RowDefault"], Background -> GrayLevel[1], StripOnInput -> 
                   False]}}, 
                GridBoxAlignment -> {
                 "Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, AutoDelete -> 
                False, GridBoxDividers -> {"Rows" -> {{None}}}, 
                GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{1.45}}},
                 GridBoxSpacings -> {"Columns" -> {{0}}}, 
                BaselinePosition -> {{1, 1}, Baseline}], "Grid"], 
              RoundingRadius -> 4, Background -> 
              GrayLevel[0.9333333333333333], 
              FrameMargins -> {{0, 0}, {-1, -1}}, FrameStyle -> {
                GrayLevel[0.8352941176470589], 
                AbsoluteThickness[1]}, BaselinePosition -> Baseline, 
              StripOnInput -> False]& )}], 
         Cell[
          StyleData["Key1"], 
          TemplateBoxOptions -> {DisplayFunction -> (FrameBox[
              TagBox[
               GridBox[{{
                  ItemBox[
                   
                   TemplateBox[{"\[ThickSpace]", #, "\[ThickSpace]"}, 
                    "RowDefault"], Background -> 
                   GrayLevel[0.9333333333333333], StripOnInput -> False], 
                  ItemBox[
                   
                   TemplateBox[{"\[ThickSpace]", #2, "\[ThickSpace]"}, 
                    "RowDefault"], Background -> GrayLevel[1], StripOnInput -> 
                   False]}}, 
                GridBoxAlignment -> {
                 "Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, AutoDelete -> 
                False, GridBoxDividers -> {
                 "ColumnsIndexed" -> {2 -> GrayLevel[0.8352941176470589]}, 
                  "Rows" -> {{None}}}, 
                GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{1.45}}},
                 GridBoxSpacings -> {"Columns" -> {{0}}}, 
                BaselinePosition -> {{1, 1}, Baseline}], "Grid"], 
              RoundingRadius -> 4, Background -> 
              GrayLevel[0.9333333333333333], 
              FrameMargins -> {{0, 0}, {-1, -1}}, FrameStyle -> {
                GrayLevel[0.8352941176470589], 
                AbsoluteThickness[1]}, BaselinePosition -> Baseline, 
              StripOnInput -> False]& )}], 
         Cell[
          StyleData["Key2"], 
          TemplateBoxOptions -> {DisplayFunction -> (FrameBox[
              TagBox[
               GridBox[{{
                  ItemBox[
                   
                   TemplateBox[{"\[ThickSpace]", #, "\[ThickSpace]"}, 
                    "RowDefault"], Background -> 
                   GrayLevel[0.9333333333333333], StripOnInput -> False], 
                  ItemBox[
                   
                   TemplateBox[{"\[ThickSpace]", #2, "\[ThickSpace]"}, 
                    "RowDefault"], Background -> 
                   GrayLevel[0.9333333333333333], StripOnInput -> False], 
                  ItemBox[
                   
                   TemplateBox[{"\[ThickSpace]", #3, "\[ThickSpace]"}, 
                    "RowDefault"], Background -> GrayLevel[1], StripOnInput -> 
                   False]}}, 
                GridBoxAlignment -> {
                 "Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, AutoDelete -> 
                False, GridBoxDividers -> {
                 "ColumnsIndexed" -> {
                   2 -> GrayLevel[0.8352941176470589], 3 -> 
                    GrayLevel[0.8352941176470589]}, "Rows" -> {{None}}}, 
                GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{1.45}}},
                 GridBoxSpacings -> {"Columns" -> {{0}}}, 
                BaselinePosition -> {{1, 1}, Baseline}], "Grid"], 
              RoundingRadius -> 4, Background -> 
              GrayLevel[0.9333333333333333], 
              FrameMargins -> {{0, 0}, {-1, -1}}, FrameStyle -> {
                GrayLevel[0.8352941176470589], 
                AbsoluteThickness[1]}, BaselinePosition -> Baseline, 
              StripOnInput -> False]& )}], 
         Cell[
          StyleData["Key3"], 
          TemplateBoxOptions -> {DisplayFunction -> (FrameBox[
              TagBox[
               GridBox[{{
                  ItemBox[
                   
                   TemplateBox[{"\[ThickSpace]", #, "\[ThickSpace]"}, 
                    "RowDefault"], Background -> 
                   GrayLevel[0.9333333333333333], StripOnInput -> False], 
                  ItemBox[
                   
                   TemplateBox[{"\[ThickSpace]", #2, "\[ThickSpace]"}, 
                    "RowDefault"], Background -> 
                   GrayLevel[0.9333333333333333], StripOnInput -> False], 
                  ItemBox[
                   
                   TemplateBox[{"\[ThickSpace]", #3, "\[ThickSpace]"}, 
                    "RowDefault"], Background -> 
                   GrayLevel[0.9333333333333333], StripOnInput -> False], 
                  ItemBox[
                   
                   TemplateBox[{"\[ThickSpace]", #4, "\[ThickSpace]"}, 
                    "RowDefault"], Background -> GrayLevel[1], StripOnInput -> 
                   False]}}, 
                GridBoxAlignment -> {
                 "Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, AutoDelete -> 
                False, GridBoxDividers -> {
                 "ColumnsIndexed" -> {
                   2 -> GrayLevel[0.8352941176470589], 3 -> 
                    GrayLevel[0.8352941176470589], 4 -> 
                    GrayLevel[0.8352941176470589]}, "Rows" -> {{None}}}, 
                GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{1.45}}},
                 GridBoxSpacings -> {"Columns" -> {{0}}}, 
                BaselinePosition -> {{1, 1}, Baseline}], "Grid"], 
              RoundingRadius -> 4, Background -> 
              GrayLevel[0.9333333333333333], 
              FrameMargins -> {{0, 0}, {-1, -1}}, FrameStyle -> {
                GrayLevel[0.8352941176470589], 
                AbsoluteThickness[1]}, BaselinePosition -> Baseline, 
              StripOnInput -> False]& )}]}, Open]], 
      Cell[
       CellGroupData[{
         Cell["Entity styles", "Subsubsection"], 
         Cell[
          StyleData["Entity"], ContextMenu -> 
          FEPrivate`FrontEndResource["ContextMenus", "Entity"], 
          AutoStyleOptions -> {"HighlightSyntaxErrors" -> False}, 
          ShowStringCharacters -> False, 
          TemplateBoxOptions -> {DisplayFunction -> (TooltipBox[
              FrameBox[
               PaneSelectorBox[{True -> GridBox[{{
                    
                    StyleBox[#, FontSize -> 0.85 Inherited, FontColor -> 
                    RGBColor[0.395437, 0.20595, 0.061158]], 
                    PaneBox[
                    StyleBox[#4, "EntityTypeLabel"], BaselinePosition -> 
                    Baseline, ContentPadding -> False]}}, 
                   GridBoxItemSize -> {
                    "Columns" -> {{All}}, "Rows" -> {{All}}}, 
                   GridBoxSpacings -> {"Columns" -> {{0.5}}, "Rows" -> {{0}}},
                    BaselinePosition -> {1, 1}], False -> 
                 PaneBox[#, 
                   BaseStyle -> {
                    FontColor -> RGBColor[0.395437, 0.20595, 0.061158]}, 
                   BaselinePosition -> Baseline]}, 
                Dynamic[
                 CurrentValue[Evaluatable]], ImageSize -> Automatic, 
                BaselinePosition -> Baseline, 
                BaseStyle -> {FontFamily -> "Helvetica"}], FrameStyle -> 
               Directive[
                 RGBColor[1., 0.504768, 0.], 
                 AbsoluteThickness[1]], Background -> 
               RGBColor[1., 0.980392, 0.921569], 
               FrameMargins -> {{5, 5}, {1.5, 1.5}}, 
               ImageMargins -> {{2, 2}, {2, 2}}, BaselinePosition -> Baseline,
                RoundingRadius -> 4, 
               BaseStyle -> {FontFamily -> "Courier"}], #3, 
              BaseStyle -> {
               ShowStringCharacters -> False, LineIndent -> 0, 
                PrivateFontOptions -> {"OperatorSubstitution" -> False}}, 
              TooltipDelay -> 0.4]& ), Editable -> False, 
            InterpretationFunction -> (#2& )}], 
         Cell[
          StyleData["EntityFrameless"], ShowStringCharacters -> False, 
          TemplateBoxOptions -> {DisplayFunction -> (TooltipBox[
              PaneSelectorBox[{True -> GridBox[{{
                    
                    StyleBox[#, FontWeight -> "Bold", FontFamily -> 
                    "Source Sans Pro", FontColor -> 
                    RGBColor[0.395437, 0.20595, 0.061158]], 
                    StyleBox[
                    RowBox[{"(", #4, ")"}], FontFamily -> "Source Sans Pro", 
                    FontWeight -> "Plain", FontColor -> GrayLevel[0.65], 
                    Selectable -> False]}}, 
                  GridBoxItemSize -> {
                   "Columns" -> {{All}}, "Rows" -> {{All}}}, 
                  GridBoxSpacings -> {"Columns" -> {{0.2}}, "Rows" -> {{0}}}, 
                  BaselinePosition -> {1, 1}], False -> 
                PaneBox[#, 
                  BaseStyle -> {
                   FontColor -> RGBColor[0.395437, 0.20595, 0.061158]}, 
                  BaselinePosition -> Baseline]}, 
               Dynamic[
                CurrentValue[Evaluatable]], ImageSize -> Automatic, 
               BaselinePosition -> Baseline, 
               BaseStyle -> {FontFamily -> "Helvetica"}], #3, 
              BaseStyle -> {
               ShowStringCharacters -> False, LineIndent -> 0, 
                PrivateFontOptions -> {
                 FontWeight -> "Plain", "OperatorSubstitution" -> False}}]& ),
             Editable -> False, InterpretationFunction -> (#2& )}], 
         Cell[
          StyleData["EntityFramelessHot"], ShowStringCharacters -> False, 
          TemplateBoxOptions -> {
           DisplayFunction -> (PaneSelectorBox[{True -> GridBox[{{
                   
                   StyleBox[#, FontWeight -> "Bold", FontFamily -> 
                    "Source Sans Pro", FontColor -> GrayLevel[0]], 
                   StyleBox[
                    RowBox[{"(", #4, ")"}], FontWeight -> "Plain", FontFamily -> 
                    "Source Sans Pro", FontColor -> GrayLevel[0], Selectable -> 
                    False]}}, 
                 GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}},
                  GridBoxSpacings -> {"Columns" -> {{0.2}}, "Rows" -> {{0}}}, 
                 BaselinePosition -> {1, 1}], False -> 
               PaneBox[#, BaseStyle -> {FontColor -> GrayLevel[0]}, 
                 BaselinePosition -> Baseline]}, 
              Dynamic[
               CurrentValue[Evaluatable]], ImageSize -> Automatic, 
              BaselinePosition -> Baseline, 
              BaseStyle -> {
               FontWeight -> "Plain", FontFamily -> "Helvetica", 
                ShowStringCharacters -> False, LineIndent -> 0, 
                PrivateFontOptions -> {"OperatorSubstitution" -> False}}]& ), 
            Editable -> False, InterpretationFunction -> (#2& )}], 
         Cell[
          StyleData["EntityClass"], ShowStringCharacters -> False, 
          TemplateBoxOptions -> {DisplayFunction -> (TooltipBox[
              FrameBox[
               PaneSelectorBox[{True -> GridBox[{{
                    PaneBox[
                    DynamicBox[
                    FEPrivate`ImportImage[
                    
                    FrontEnd`ToFileName[{"Typeset", "Entity"}, 
                    "EntityClass.png"]], ImageSizeCache -> {8., {2., 6.}}], 
                    BaselinePosition -> Bottom], 
                    
                    StyleBox[#, FontWeight -> "Bold", FontFamily -> 
                    "Source Sans Pro", FontColor -> 
                    RGBColor[0.395437, 0.20595, 0.061158]], 
                    StyleBox[
                    RowBox[{"(", #4, ")"}], FontWeight -> "Plain", FontFamily -> 
                    "Source Sans Pro", FontColor -> GrayLevel[0.65], 
                    Selectable -> False]}}, 
                   GridBoxItemSize -> {
                    "Columns" -> {{All}}, "Rows" -> {{All}}}, 
                   GridBoxSpacings -> {"Columns" -> {{0.2}}, "Rows" -> {{0}}},
                    BaselinePosition -> {1, 2}], False -> GridBox[{{
                    PaneBox[
                    DynamicBox[
                    FEPrivate`ImportImage[
                    
                    FrontEnd`ToFileName[{"Typeset", "Entity"}, 
                    "EntityClass.png"]], ImageSizeCache -> {8., {2., 6.}}], 
                    BaselinePosition -> Bottom], 
                    
                    StyleBox[#, FontColor -> 
                    RGBColor[0.395437, 0.20595, 0.061158]]}}, 
                   GridBoxItemSize -> {
                    "Columns" -> {{All}}, "Rows" -> {{All}}}, 
                   GridBoxSpacings -> {"Columns" -> {{0.2}}, "Rows" -> {{0}}},
                    BaselinePosition -> {1, 2}]}, 
                Dynamic[
                 CurrentValue[Evaluatable]], ImageSize -> Automatic, 
                BaselinePosition -> Baseline, 
                BaseStyle -> {FontFamily -> "Helvetica"}], FrameStyle -> 
               RGBColor[1., 0.504768, 0.], Background -> 
               RGBColor[1., 0.980392, 0.921569], 
               FrameMargins -> {{5, 5}, {1.5, 1.5}}, 
               ImageMargins -> {{2, 2}, {2, 2}}, BaselinePosition -> Baseline,
                RoundingRadius -> 4], #3, 
              BaseStyle -> {
               FontWeight -> "Plain", ShowStringCharacters -> False, 
                LineIndent -> 0, 
                PrivateFontOptions -> {"OperatorSubstitution" -> False}}]& ), 
            Editable -> False, InterpretationFunction -> (#2& )}], 
         Cell[
          StyleData["EntityClassFrameless"], ShowStringCharacters -> False, 
          TemplateBoxOptions -> {DisplayFunction -> (TooltipBox[
              PaneSelectorBox[{True -> GridBox[{{
                    PaneBox[
                    DynamicBox[
                    FEPrivate`ImportImage[
                    
                    FrontEnd`ToFileName[{"Typeset", "Entity"}, 
                    "EntityClass.png"]], ImageSizeCache -> {8., {2., 6.}}], 
                    BaselinePosition -> Bottom], 
                    
                    StyleBox[#, FontWeight -> "Bold", FontFamily -> 
                    "Source Sans Pro", FontColor -> 
                    RGBColor[0.395437, 0.20595, 0.061158]], 
                    StyleBox[
                    RowBox[{"(", #4, ")"}], FontWeight -> "Plain", FontFamily -> 
                    "Source Sans Pro", FontColor -> GrayLevel[0.65], 
                    Selectable -> False]}}, 
                  GridBoxItemSize -> {
                   "Columns" -> {{All}}, "Rows" -> {{All}}}, 
                  GridBoxSpacings -> {"Columns" -> {{0.2}}, "Rows" -> {{0}}}, 
                  BaselinePosition -> {1, 2}], False -> GridBox[{{
                    PaneBox[
                    DynamicBox[
                    FEPrivate`ImportImage[
                    
                    FrontEnd`ToFileName[{"Typeset", "Entity"}, 
                    "EntityClass.png"]], ImageSizeCache -> {8., {2., 6.}}], 
                    BaselinePosition -> Bottom], 
                    
                    StyleBox[#, FontColor -> 
                    RGBColor[0.395437, 0.20595, 0.061158]]}}, 
                  GridBoxItemSize -> {
                   "Columns" -> {{All}}, "Rows" -> {{All}}}, 
                  GridBoxSpacings -> {"Columns" -> {{0.2}}, "Rows" -> {{0}}}, 
                  BaselinePosition -> {1, 2}]}, 
               Dynamic[
                CurrentValue[Evaluatable]], ImageSize -> Automatic, 
               BaselinePosition -> Baseline, 
               BaseStyle -> {FontFamily -> "Helvetica"}], #3, 
              BaseStyle -> {
               FontWeight -> "Plain", ShowStringCharacters -> False, 
                LineIndent -> 0, 
                PrivateFontOptions -> {"OperatorSubstitution" -> False}}]& ), 
            Editable -> False, InterpretationFunction -> (#2& )}], 
         Cell[
          StyleData["EntityClassFramelessHot"], ShowStringCharacters -> False,
           TemplateBoxOptions -> {
           DisplayFunction -> (PaneSelectorBox[{True -> GridBox[{{
                   PaneBox[
                    DynamicBox[
                    FEPrivate`ImportImage[
                    
                    FrontEnd`ToFileName[{"Typeset", "Entity"}, 
                    "EntityClass.png"]], ImageSizeCache -> {8., {2., 6.}}], 
                    BaselinePosition -> Bottom], 
                   
                   StyleBox[#, FontWeight -> "Bold", FontFamily -> 
                    "Source Sans Pro", FontColor -> GrayLevel[0]], 
                   StyleBox[
                    RowBox[{"(", #4, ")"}], FontWeight -> "Plain", FontFamily -> 
                    "Source Sans Pro", FontColor -> GrayLevel[0], Selectable -> 
                    False]}}, 
                 GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}},
                  GridBoxSpacings -> {"Columns" -> {{0.2}}, "Rows" -> {{0}}}, 
                 BaselinePosition -> {1, 2}], False -> GridBox[{{
                   PaneBox[
                    DynamicBox[
                    FEPrivate`ImportImage[
                    
                    FrontEnd`ToFileName[{"Typeset", "Entity"}, 
                    "EntityClass.png"]], ImageSizeCache -> {8., {2., 6.}}], 
                    BaselinePosition -> Bottom], 
                   StyleBox[#, FontColor -> GrayLevel[0]]}}, 
                 GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}},
                  GridBoxSpacings -> {"Columns" -> {{0.2}}, "Rows" -> {{0}}}, 
                 BaselinePosition -> {1, 2}]}, 
              Dynamic[
               CurrentValue[Evaluatable]], ImageSize -> Automatic, 
              BaselinePosition -> Baseline, 
              BaseStyle -> {
               FontWeight -> "Plain", FontFamily -> "Helvetica", 
                ShowStringCharacters -> False, LineIndent -> 0, 
                PrivateFontOptions -> {"OperatorSubstitution" -> False}}]& ), 
            Editable -> False, InterpretationFunction -> (#2& )}], 
         Cell[
          StyleData["EntityProperty"], ContextMenu -> 
          FEPrivate`FrontEndResource["ContextMenus", "Entity"], 
          AutoStyleOptions -> {"HighlightSyntaxErrors" -> False}, 
          ShowStringCharacters -> False, 
          TemplateBoxOptions -> {DisplayFunction -> (TooltipBox[
              FrameBox[
               
               PaneBox[#, 
                BaseStyle -> {
                 FontSlant -> Italic, FontWeight -> Plain, FontFamily -> 
                  "Helvetica", FontSize -> 0.85 Inherited, FontColor -> 
                  RGBColor[0.395437, 0.20595, 0.061158]}, BaselinePosition -> 
                Baseline], FrameStyle -> Directive[
                 RGBColor[0.94227, 0.703639, 0.033387], 
                 AbsoluteThickness[1]], Background -> 
               RGBColor[1., 0.980392, 0.921569], 
               FrameMargins -> {{5, 5}, {1.5, 1.5}}, 
               ImageMargins -> {{2, 2}, {2, 2}}, BaselinePosition -> Baseline,
                RoundingRadius -> 4, 
               BaseStyle -> {FontFamily -> "Courier"}], #3, 
              BaseStyle -> {
               ShowStringCharacters -> False, LineIndent -> 0, 
                PrivateFontOptions -> {"OperatorSubstitution" -> False}}, 
              TooltipDelay -> 0.4]& ), Editable -> False, 
            InterpretationFunction -> (#2& )}], 
         Cell[
          StyleData["EntityPropertyClass"], ShowStringCharacters -> False, 
          TemplateBoxOptions -> {DisplayFunction -> (TooltipBox[
              FrameBox[
               PaneBox[
                GridBox[{{
                   PaneBox[
                    DynamicBox[
                    FEPrivate`ImportImage[
                    
                    FrontEnd`ToFileName[{"Typeset", "Entity"}, 
                    "EntityClass.png"]], ImageSizeCache -> {8., {2., 6.}}], 
                    BaselinePosition -> Bottom], #}}, 
                 GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}},
                  GridBoxSpacings -> {"Columns" -> {{0.2}}, "Rows" -> {{0}}}, 
                 BaselinePosition -> {1, 2}], 
                BaseStyle -> {
                 FontSlant -> Italic, FontWeight -> Plain, FontFamily -> 
                  "Source Sans Pro", FontColor -> 
                  RGBColor[0.395437, 0.20595, 0.061158]}, BaselinePosition -> 
                Baseline], FrameStyle -> 
               RGBColor[0.94227, 0.703639, 0.033387], Background -> 
               RGBColor[1., 0.980392, 0.921569], 
               FrameMargins -> {{5, 5}, {1.5, 1.5}}, 
               ImageMargins -> {{2, 2}, {2, 2}}, BaselinePosition -> Baseline,
                RoundingRadius -> 4], #3, 
              BaseStyle -> {
               FontWeight -> "Plain", ShowStringCharacters -> False, 
                LineIndent -> 0, 
                PrivateFontOptions -> {"OperatorSubstitution" -> False}}]& ), 
            Editable -> False, InterpretationFunction -> (#2& )}], 
         Cell[
          StyleData["EntityValue"], ShowStringCharacters -> False, 
          TemplateBoxOptions -> {DisplayFunction -> (FrameBox[
              
              GridBox[{{#, #2}}, GridBoxAlignment -> {"Rows" -> {{Baseline}}},
                GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, 
               GridBoxSpacings -> {"Columns" -> {{0}}}, 
               BaselinePosition -> {1, 1}], Background -> 
              RGBColor[{0.9921568627450981, 0.9529411764705882, 
                0.8509803921568627}], BaselinePosition -> Baseline, 
              FrameMargins -> {{1, 1}, {1, 1}}, ImageMargins -> 2, FrameStyle -> 
              RGBColor[{0.9333333333333333, 0.8470588235294118, 
                0.5529411764705883}], RoundingRadius -> 5]& )}]}, Open]]}, 
     Open]], 
   Cell[
    CellGroupData[{
      Cell["Template Boxes", "Section"], 
      Cell[
       StyleData["BannerTemplate"], 
       TemplateBoxOptions -> {DisplayFunction :> (TagBox[
           GridBox[{{
              Slot["label"], 
              Slot["previous"], 
              TemplateBox[{10}, "Spacer1"], 
              Slot["next"], 
              TemplateBox[{10}, "Spacer1"], 
              Slot["menu"]}}, 
            GridBoxAlignment -> {
             "Columns" -> {{Left, Right}}, "Rows" -> {{Center}}}, AutoDelete -> 
            False, GridBoxFrame -> {"Columns" -> None, "Rows" -> {{None}}}, 
            GridBoxItemSize -> {
             "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, 
            GridBoxSpacings -> {"Columns" -> {{0}}, "Rows" -> {{0}}}], 
           "Grid"]& )}], 
      Cell[
       StyleData["BannerHeadingTitleTemplate"], 
       TemplateBoxOptions -> {DisplayFunction :> (PanelBox[
           TagBox[
            GridBox[{{
               StyleBox[
                Slot["heading"], "BannerHeading", StripOnInput -> False]}, {
               Slot["title"]}}, GridBoxAlignment -> {"Columns" -> {{Left}}}, 
             DefaultBaseStyle -> "Column", 
             GridBoxItemSize -> {
              "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, 
             GridBoxSpacings -> {
              "Columns" -> {{Automatic}}, "Rows" -> {{0.2}}}], "Column"], 
           Appearance -> None, ContentPadding -> False, BaseStyle -> "Banner",
            FrameMargins -> {{18, 20}, {6, 6}}, 
           ImageSize -> {Full, Automatic}]& )}], 
      Cell[
       StyleData["BannerButtonTemplate"], 
       LightDarkAutoColorRules -> <|"MarkedColors" -> True|>, 
       TemplateBoxOptions -> {DisplayFunction :> (ButtonBox[
           
           PanelBox[#, ContentPadding -> False, 
            FrameMargins -> {{10, 10}, {7, 7}}, Appearance -> None, BaseStyle -> 
            Dynamic[
              FEPrivate`If[
               CurrentValue["MouseOver"], {
               Background -> 
                RGBColor[
                 0.08808648687787803, 0.3397116701677621, 
                  0.5610151372105242]}, "Banner"], Evaluator -> "System"]], 
           ButtonFunction -> #2, Appearance -> None, 
           FrameMargins -> {{0, 0}, {-1, -1}}, Enabled -> True, Evaluator -> 
           Automatic, Method -> "Preemptive"]& )}], 
      Cell[
       StyleData["BannerLeftArrowTemplate"], 
       TemplateBoxOptions -> {DisplayFunction -> (GraphicsBox[{
            CapForm["Round"], 
            JoinForm["Round"], 
            Thickness[0.1], 
            GrayLevel[1], 
            LineBox[{{25, 180}, {-25, 0}, {25, -180}}]}, {
           Method -> {"ShrinkWrap" -> True}, ImageSize -> 12, 
            PlotRange -> {{-75, 100}, {-275, 400}}, AspectRatio -> 1}]& )}], 
      Cell[
       StyleData["BannerRightArrowTemplate"], 
       TemplateBoxOptions -> {DisplayFunction -> (GraphicsBox[{
            CapForm["Round"], 
            JoinForm["Round"], 
            Thickness[0.1], 
            GrayLevel[1], 
            LineBox[{{-25, -180}, {25, 0}, {-25, 180}}]}, {
           Method -> {"ShrinkWrap" -> True}, ImageSize -> 12, 
            PlotRange -> {{-75, 100}, {-275, 400}}, AspectRatio -> 1}]& )}], 
      Cell[
       StyleData["PrettyTooltipTemplate"], 
       TemplateBoxOptions -> {DisplayFunction -> (TooltipBox[#, 
           FrameBox[
            
            StyleBox[#2, FontFamily -> "Source Sans Pro", FontColor -> 
             RGBColor[0.53725, 0.53725, 0.53725], FontSize -> 12, FontWeight -> 
             "Plain", FontTracking -> "Plain", StripOnInput -> False], 
            Background -> RGBColor[0.96078, 0.96078, 0.96078], FrameStyle -> 
            RGBColor[0.89804, 0.89804, 0.89804], FrameMargins -> 8, 
            StripOnInput -> False], TooltipDelay -> 0.1, 
           TooltipStyle -> {Background -> None, CellFrame -> 0}]& )}], 
      Cell[
       StyleData["BannerMenuTemplate"], 
       LightDarkAutoColorRules -> <|"MarkedColors" -> True|>, 
       TemplateBoxOptions -> {DisplayFunction :> (ActionMenuBox[
           TagBox[
            PanelBox[
             GraphicsBox[{
               CapForm["Round"], 
               GrayLevel[1], 
               AbsoluteThickness[3.2], 
               LineBox[{{0, 0}, {2, 0}}], 
               LineBox[{{0, 1}, {2, 1}}], 
               LineBox[{{0, 2}, {2, 2}}]}, ImageSize -> {27, Automatic}, 
              AspectRatio -> 0.62], ContentPadding -> False, 
             FrameMargins -> {{10, 10}, {7, 7}}, Appearance -> None, 
             BaseStyle -> Dynamic[
               FEPrivate`If[
                CurrentValue["MouseOver"], {
                Background -> 
                 RGBColor[
                  0.08808648687787803, 0.3397116701677621, 
                   0.5610151372105242]}, "Banner"], Evaluator -> "System"]], 
            MouseAppearanceTag["LinkHand"]], #, Appearance -> None]& )}], 
      Cell[
       StyleData["BannerCitationOpenerTemplate"], 
       TemplateBoxOptions -> {DisplayFunction :> (ButtonBox[
           TagBox[
            GridBox[{{
               Slot["label"], 
               PaneSelectorBox[{True -> GraphicsBox[{
                    CapForm["Round"], 
                    JoinForm["Round"], 
                    Thickness[0.18], 
                    GrayLevel[0.7], 
                    LineBox[{{-180, 25}, {0, -25}, {180, 25}}]}, {
                   Method -> {"ShrinkWrap" -> True}, ImageSize -> 11, 
                    PlotRange -> {{-275, 400}, {-75, 100}}, AspectRatio -> 
                    1}], False -> GraphicsBox[{
                    CapForm["Round"], 
                    JoinForm["Round"], 
                    Thickness[0.18], 
                    GrayLevel[0.7], 
                    LineBox[{{-25, -180}, {25, 0}, {-25, 180}}]}, {
                   Method -> {"ShrinkWrap" -> True}, ImageSize -> 8, 
                    PlotRange -> {{-75, 100}, {-475, 200}}, AspectRatio -> 
                    1}]}, 
                Dynamic[
                 CurrentValue[
                  EvaluationNotebook[], {
                  TaggingRules, "eTextbook", "CitationOpener"}]]]}}, 
             GridBoxAlignment -> {"Rows" -> {{Baseline}}}, AutoDelete -> 
             False, GridBoxItemSize -> {
              "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Grid"], 
           ButtonFunction :> (CurrentValue[
              EvaluationNotebook[], {
              TaggingRules, "eTextbook", "CitationOpener"}] = Not[
              TrueQ[
               CurrentValue[
                EvaluationNotebook[], {
                TaggingRules, "eTextbook", "CitationOpener"}]]]), Appearance -> 
           None, Evaluator -> Automatic, Method -> "Preemptive", BaseStyle -> 
           "BannerCitationOpenerLabel"]& )}], 
      Cell[
       StyleData["CitationClickToCopy2"], 
       TemplateBoxOptions -> {DisplayFunction -> (TagBox[
           DynamicModuleBox[{Typeset`boxobj$$, Typeset`cellobj$$}, 
            TagBox[
             TagBox[
              ButtonBox[
               
               TagBox[#, BoxForm`Undeploy, 
                DefaultBaseStyle -> {Deployed -> False}], ButtonFunction :> 
               FrontEndExecute[{
                  CopyToClipboard[
                   Cell[
                    TextData[#2], "Text", 
                    LinebreakAdjustments -> {0.85, 2, 10, 0, 1}, 
                    DefaultFormatType -> DefaultTextFormatType, 
                    TextClipboardType -> "PlainText", StripStyleOnPaste -> 
                    True, PageBreakWithin -> False, GroupPageBreakWithin -> 
                    False, LanguageCategory -> None, 
                    HyphenationOptions -> {"HyphenationCharacter" -> ""}]], 
                  NotebookDelete[Typeset`cellobj$$], 
                  FrontEnd`AttachCell[Typeset`boxobj$$, 
                   Cell[
                    BoxData[
                    TemplateBox[{"Copied"}, "ClickToCopyTooltip"]]], {
                   1, {Center, Bottom}}, {Center, Top}, 
                   "ClosingActions" -> {"ParentChanged", "MouseExit"}]}], 
               Evaluator -> None, 
               Appearance -> {
                "Default" -> None, "Hover" -> 
                 FrontEnd`FileName[{"Typeset", "ClickToCopy"}, "Hover.9.png"],
                  "Pressed" -> 
                 FrontEnd`FileName[{"Typeset", "ClickToCopy"}, 
                   "Pressed.9.png"]}, BaseStyle -> {}, DefaultBaseStyle -> {},
                BaselinePosition -> Baseline, FrameMargins -> 2, Method -> 
               "Preemptive"], 
              
              EventHandlerTag[{
               "MouseEntered" :> (Typeset`cellobj$$ = MathLink`CallFrontEnd[
                   FrontEnd`AttachCell[Typeset`boxobj$$, 
                    Cell[
                    BoxData[
                    TemplateBox[{"Copy"}, "ClickToCopyTooltip"]]], {
                    1, {Center, Bottom}}, {Center, Top}, 
                    "ClosingActions" -> {"ParentChanged"}]]), "MouseExited" :> 
                NotebookDelete[Typeset`cellobj$$], PassEventsDown -> True, 
                Method -> "Preemptive", PassEventsUp -> True}]], 
             MouseAppearanceTag["LinkHand"]], 
            Initialization :> (Typeset`boxobj$$ = EvaluationBox[]), 
            DynamicModuleValues :> {}, 
            UnsavedVariables :> {Typeset`boxobj$$, Typeset`cellobj$$}, 
            BaseStyle -> {Editable -> False}], Deploy, DefaultBaseStyle -> 
           "Deploy"]& ), 
         InterpretationFunction -> (RowBox[{"ClickToCopy", "[", 
            RowBox[{#, ",", #2}], "]"}]& ), Tooltip -> None}], 
      Cell[
       StyleData["BannerPreReleaseTemplate"], 
       TemplateBoxOptions -> {DisplayFunction :> (TagBox[
           GridBox[{{
              PanelBox[
               StyleBox[
                Slot["author"], FontWeight -> "SemiBold"], BaseStyle -> 
               "BannerPreReleaseLabel", Appearance -> None, ContentPadding -> 
               False, FrameMargins -> {{18, 20}, {6, 6}}, 
               ImageSize -> {Full, Automatic}], 
              TemplateBox[{"Provide Feedback \[RightGuillemet]", {
                 URL[
                  Slot["url"]], None}, 
                Slot["url"], "BannerPreReleaseLabel", {"HyperlinkActive"}, {
                Appearance -> None}}, "HyperlinkTemplate"]}}, 
            GridBoxAlignment -> {
             "Columns" -> {{Left, Right}}, "Rows" -> {{Center}}}, AutoDelete -> 
            False, GridBoxFrame -> {"Columns" -> None, "Rows" -> {{None}}}, 
            GridBoxItemSize -> {
             "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, 
            GridBoxSpacings -> {"Columns" -> {{0}}, "Rows" -> {{0}}}], 
           "Grid"]& )}]}, Open]]}, Visible -> False, FrontEndVersion -> 
  "14.3 for Mac OS X ARM (64-bit) (July 1, 2025)", StyleDefinitions -> 
  "PrivateStylesheetFormatting.nb"],
ETextbookTools`CreateNotebookCollection`Private`FileChangeProtection -> None,
ExpressionUUID->"e1dfdf71-9e21-4f82-8da2-9c32309e5240"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[648, 23, 226, 7, 108, "Section",ExpressionUUID->"aba84274-cae8-4d31-a68e-7cd590a35450",
 CounterAssignments->{{"Section", 3}},
 CellID->1383992936],
Cell[877, 32, 574, 12, 60, "Text",ExpressionUUID->"431f6b64-6b41-48a4-90c8-32f8d3933a9c",
 CellID->1980475242],
Cell[1454, 46, 522, 9, 130, "Text",ExpressionUUID->"65f17eeb-da0a-4ea4-9a74-4039423f498e",
 CellID->1223975305],
Cell[1979, 57, 352, 6, 60, "Text",ExpressionUUID->"88ab9ba2-299d-4e8c-99b2-aa54db8127f8",
 CellID->983075642],
Cell[2334, 65, 774, 13, 83, "Text",ExpressionUUID->"6299ef7d-d67f-445a-bfda-6272fb4e7722",
 CellID->1025122044],
Cell[3111, 80, 673, 13, 83, "Text",ExpressionUUID->"726d9eb2-ba1b-4cb1-aa5a-eb8ce42aeff4",
 CellID->2112377274],
Cell[3787, 95, 727, 12, 130, "Text",ExpressionUUID->"a109a645-464b-4a5c-9f44-7c40c5c69865",
 CellID->802375479],
Cell[4517, 109, 693, 10, 59, "Text",ExpressionUUID->"f760a491-e37d-4f70-a123-445de67eb9b5",
 CellID->525191914],
Cell[CellGroupData[{
Cell[5235, 123, 662, 9, 56, "Subsection",ExpressionUUID->"d577cabd-46ec-4803-9d6a-f6b5b53a4df6",
 CellID->1542400032],
Cell[5900, 134, 388, 5, 36, "Text",ExpressionUUID->"8b97fd45-56d6-469e-a8b6-a0531d89dd78",
 CellID->842234887],
Cell[6291, 141, 418, 7, 59, "Text",ExpressionUUID->"0183d05e-d071-44b0-9a17-78617ecbe79e",
 CellID->1377028432],
Cell[6712, 150, 236, 3, 36, "Text",ExpressionUUID->"b53a84c6-9628-4ea9-b318-7802593a35f2",
 CellID->285592905],
Cell[6951, 155, 331, 4, 36, "Text",ExpressionUUID->"0eb20aa9-b09f-40fb-9db8-2ce779570793",
 CellID->334003290],
Cell[7285, 161, 185, 4, 32, "DisplayFormula",ExpressionUUID->"c7918e49-9270-4708-aefb-7b0dc97241c2",
 CellID->57689067],
Cell[7473, 167, 199, 4, 32, "DisplayFormula",ExpressionUUID->"6c6d688f-6804-4142-9b06-8b61337cfec0",
 CellID->1909381740],
Cell[7675, 173, 232, 5, 32, "DisplayFormula",ExpressionUUID->"72d5dfc5-8e77-429c-abed-375c240c96b4",
 CellID->1130767059],
Cell[7910, 180, 165, 3, 32, "DisplayFormula",ExpressionUUID->"59546147-9cf7-4f24-9d6f-cddaec66caf2",
 CellID->1719168310],
Cell[8078, 185, 916, 14, 59, "Text",ExpressionUUID->"a7230b2c-3a60-462a-83e6-cca0f2acdd8f",
 CellID->214481230],
Cell[8997, 201, 231, 5, 32, "DisplayFormula",ExpressionUUID->"40b2cf7d-79bc-40cf-a8ff-f3e4a3051226",
 CellID->297981691],
Cell[9231, 208, 231, 5, 32, "DisplayFormula",ExpressionUUID->"6088ad3a-4182-4454-9080-2d7981baa62f",
 CellID->246415833],
Cell[9465, 215, 232, 5, 32, "DisplayFormula",ExpressionUUID->"e6459dd6-70a4-407c-9115-4e7d239bf9f6",
 CellID->1162433815],
Cell[9700, 222, 763, 12, 83, "Text",ExpressionUUID->"197b7eb2-c365-4c9a-9384-1d9b596769da",
 CellID->933304272],
Cell[10466, 236, 250, 6, 32, "DisplayFormula",ExpressionUUID->"51f8b649-b922-46fa-8737-b68ad040c1c6",
 CellID->673185976]
}, Open  ]],
Cell[CellGroupData[{
Cell[10753, 247, 513, 7, 56, "Subsection",ExpressionUUID->"f1a7ee64-bdd9-4600-b39c-f11be83c4c1c",
 CellID->323327131],
Cell[11269, 256, 595, 10, 83, "Text",ExpressionUUID->"5acfcdb1-a2ce-4a8b-9d95-301575c831b3",
 CellID->1025463807],
Cell[11867, 268, 754, 13, 83, "Text",ExpressionUUID->"bb29dca0-9248-413f-b358-2a4f5600ab74",
 CellID->1335604159],
Cell[12624, 283, 599, 12, 36, "Text",ExpressionUUID->"d379a323-a521-4726-a598-713c5f78c71a",
 CellID->160259201],
Cell[13226, 297, 382, 10, 33, "DisplayFormula",ExpressionUUID->"670e45ea-b165-4c60-860e-2686d8548462",
 CellID->98567012],
Cell[13611, 309, 284, 8, 36, "Text",ExpressionUUID->"b5fb5914-dba0-46f9-b12e-c17dbc6357e3",
 CellID->780121836],
Cell[13898, 319, 557, 13, 106, "Text",ExpressionUUID->"6d222f70-ad0f-4df7-8cbb-135e1a6552fa",
 CellID->1120032717],
Cell[14458, 334, 286, 8, 32, "DisplayFormula",ExpressionUUID->"f408a987-844a-4927-8c16-756c7ec1f651",
 CellID->1285747781]
}, Open  ]],
Cell[CellGroupData[{
Cell[14781, 347, 654, 9, 56, "Subsection",ExpressionUUID->"771370dc-0ad1-4aae-b3ad-32cca02f48e3",
 CellID->2142680734],
Cell[15438, 358, 380, 5, 36, "Text",ExpressionUUID->"7c3161bd-d5f8-4b9e-a9e4-3aa5c564294c",
 CellID->881819795],
Cell[15821, 365, 418, 7, 59, "Text",ExpressionUUID->"0f572605-edd4-4524-95f6-4ece577bd4f4",
 CellID->960646618],
Cell[16242, 374, 165, 3, 32, "DisplayFormula",ExpressionUUID->"b9dd5c18-4982-4b85-8e0a-958a326a3883",
 CellID->2042058743],
Cell[16410, 379, 166, 3, 32, "DisplayFormula",ExpressionUUID->"ec3651ce-29f6-4f89-bc32-97c9dc360631",
 CellID->848621744],
Cell[16579, 384, 388, 5, 36, "Text",ExpressionUUID->"dedf254f-2e43-404c-af40-7a33eb89f6d0",
 CellID->664558155],
Cell[16970, 391, 188, 4, 32, "DisplayFormula",ExpressionUUID->"34ae9ec2-a661-4599-b052-a31cc88124d6",
 CellID->1776114313],
Cell[17161, 397, 214, 5, 32, "DisplayFormula",ExpressionUUID->"ab8ee790-5d91-4ff7-b443-4f0d19a7b50c",
 CellID->1130565291],
Cell[17378, 404, 216, 6, 32, "DisplayFormula",ExpressionUUID->"05a08401-f29e-45fa-ace3-982058d2f900",
 CellID->475537181],
Cell[17597, 412, 218, 6, 32, "DisplayFormula",ExpressionUUID->"82470124-087e-4e1a-832c-126af6743ca4",
 CellID->1566750580],
Cell[17818, 420, 252, 6, 32, "DisplayFormula",ExpressionUUID->"78064d91-6a20-4859-820f-03931d7a2134",
 CellID->2080002193],
Cell[18073, 428, 224, 5, 32, "DisplayFormula",ExpressionUUID->"15395355-f2ac-40c9-b4b4-bfd5a5dfcf48",
 CellID->1858051885]
}, Open  ]],
Cell[CellGroupData[{
Cell[18334, 438, 247, 3, 56, "Subsection",ExpressionUUID->"a7281f9e-bd38-4e8b-8d2c-5fe6e8792636",
 CellID->1581852389],
Cell[18584, 443, 315, 7, 59, "Text",ExpressionUUID->"49f07b95-b228-40a3-85ad-c575e26adc93",
 CellID->345552063],
Cell[18902, 452, 171, 2, 28, "Input",ExpressionUUID->"21ce2761-a887-44a1-a6fc-031f33432c28",
 CellID->897741353],
Cell[CellGroupData[{
Cell[19098, 458, 552, 14, 25, "Input",ExpressionUUID->"cc5ff134-f3ff-406b-94a6-ceda0da2a409",
 CellID->1607086438],
Cell[19653, 474, 5003, 123, 240, "Output",ExpressionUUID->"f1c18c41-e7be-44b5-a588-728babf8d392",
 CellID->1675591919]
}, {2}]],
Cell[24668, 600, 859, 23, 83, "Text",ExpressionUUID->"8c5517ee-d2ed-445c-b693-cb7b157941b6",
 CellID->604237601],
Cell[25530, 625, 488, 13, 59, "Text",ExpressionUUID->"3dd53632-f5b9-40f0-b72a-e2dd7f164f9e",
 CellID->1070036956]
}, Open  ]],
Cell[CellGroupData[{
Cell[26055, 643, 682, 9, 56, "Subsection",ExpressionUUID->"eb2456e1-dfb2-4faf-a3e8-6356c042df54",
 CellID->399343764],
Cell[26740, 654, 279, 7, 36, "Text",ExpressionUUID->"acec063d-2d5c-4ae6-8530-8de5a3873249",
 CellID->884244673],
Cell[27022, 663, 252, 6, 32, "DisplayFormula",ExpressionUUID->"9b580d62-6053-4424-82af-1b7adc352969",
 CellID->1426267429],
Cell[27277, 671, 334, 4, 36, "Text",ExpressionUUID->"cf5f7ce1-7bd3-42a4-98c4-3d76ada358b4",
 CellID->995997141],
Cell[CellGroupData[{
Cell[27636, 679, 264, 6, 34, "Item",ExpressionUUID->"fc89f60b-c88a-4a0f-b86f-33ebbae16739",
 CellID->88920422],
Cell[27903, 687, 343, 9, 34, "Item",ExpressionUUID->"aff0d6d4-5fa1-4004-8d07-75e615dd7afd",
 CellID->1984397889],
Cell[28249, 698, 294, 7, 34, "Item",ExpressionUUID->"a4c6fbd6-a2e6-47c2-b8fb-ba2b8b9bac12",
 CellID->1354282513]
}, Open  ]],
Cell[28558, 708, 394, 7, 59, "Text",ExpressionUUID->"3eee7785-87bb-4a0b-b19d-d0af07d11ee6",
 CellID->247021438]
}, Open  ]],
Cell[CellGroupData[{
Cell[28989, 720, 614, 8, 56, "Subsection",ExpressionUUID->"5e0ca36d-7a01-4ab7-b299-2df8e409709c",
 CellID->1242242571],
Cell[29606, 730, 404, 11, 59, "Text",ExpressionUUID->"b51278b4-83a8-4f6b-b295-15cbf3f8538f",
 CellID->534874663],
Cell[30013, 743, 567, 18, 37, "Text",ExpressionUUID->"307091de-d6dc-41a7-9926-4e14237f06fc",
 CellID->393468604],
Cell[30583, 763, 383, 11, 33, "DisplayFormula",ExpressionUUID->"e55475fa-a625-46dc-886c-982bfde55c17",
 CellID->1744449662],
Cell[30969, 776, 429, 12, 36, "Text",ExpressionUUID->"25670f9a-5a41-433f-a661-cd227e293494",
 CellID->150484525],
Cell[31401, 790, 415, 12, 36, "Text",ExpressionUUID->"f9e86950-2a73-4f89-aec7-7ec5a944e9c8",
 CellID->1605159156],
Cell[CellGroupData[{
Cell[31841, 806, 1712, 43, 27, "Input",ExpressionUUID->"1bb582be-582d-49d1-ab0f-a32b9bd5de10",
 CellID->1220361278],
Cell[33556, 851, 751, 19, 63, "Output",ExpressionUUID->"0e766754-17e7-4cbe-87a3-44088279a363",
 CellID->858094877]
}, {2}]],
Cell[34319, 873, 362, 9, 36, "Text",ExpressionUUID->"74a38599-c47b-466a-8e0a-5f811c92aa02",
 CellID->523315141],
Cell[CellGroupData[{
Cell[34706, 886, 586, 14, 25, "Input",ExpressionUUID->"996ccd72-3e08-4108-92c6-b271a63cbb05",
 CellID->1635199970],
Cell[35295, 902, 4621, 117, 234, "Output",ExpressionUUID->"df401834-6bc2-44e8-b27a-4e376ba9c6ac",
 CellID->1360134510]
}, {2}]]
}, Open  ]],
Cell[CellGroupData[{
Cell[39962, 1025, 661, 9, 56, "Subsection",ExpressionUUID->"c9c90112-24b7-482d-b174-3171aceb1d5a",
 CellID->981804752],
Cell[40626, 1036, 400, 11, 59, "Text",ExpressionUUID->"041886e3-cfc6-410e-8c92-1a2f291b42fb",
 CellID->2075606963],
Cell[41029, 1049, 749, 24, 37, "Text",ExpressionUUID->"d6508ebc-d568-4a87-8daf-ae7bab118256",
 CellID->409894930],
Cell[41781, 1075, 518, 17, 32, "DisplayFormula",ExpressionUUID->"4955f0ba-2a6d-48ea-b598-fb1fd4ef6d46",
 CellID->1730623447],
Cell[42302, 1094, 228, 5, 32, "DisplayFormula",ExpressionUUID->"8dcd7fcf-6c7d-4163-b04d-62df3017ac07",
 CellID->1590926560],
Cell[42533, 1101, 424, 12, 36, "Text",ExpressionUUID->"5a5acc43-22fa-4a3d-8db6-c59b791783d8",
 CellID->59849103],
Cell[42960, 1115, 540, 16, 36, "Text",ExpressionUUID->"5841051a-73aa-43b3-bf34-56cf38406680",
 CellID->861527325],
Cell[CellGroupData[{
Cell[43525, 1135, 1830, 45, 27, "Input",ExpressionUUID->"0d42e194-b9a8-47ff-8510-6070e0dc6b9b",
 CellID->1355003201],
Cell[45358, 1182, 956, 21, 63, "Output",ExpressionUUID->"c7c99c40-4ae7-4612-8e9e-cf6e5a71b79d",
 CellID->56005488]
}, {2}]],
Cell[46326, 1206, 361, 9, 36, "Text",ExpressionUUID->"5c778cc6-a4b3-462f-88ba-ccf4b81b1939",
 CellID->2118385468],
Cell[CellGroupData[{
Cell[46712, 1219, 576, 14, 25, "Input",ExpressionUUID->"bcc4f2b4-ff43-4ed2-a617-ea56101d308b",
 CellID->72970917],
Cell[47291, 1235, 4601, 117, 234, "Output",ExpressionUUID->"eb54da94-c688-4de6-a349-8173000cafb1",
 CellID->1850105899]
}, {2}]]
}, Open  ]],
Cell[CellGroupData[{
Cell[51938, 1358, 679, 9, 56, "Subsection",ExpressionUUID->"369b4c3e-184a-4c97-9079-1791d506fc7d",
 CellID->779360166],
Cell[52620, 1369, 468, 12, 59, "Text",ExpressionUUID->"abcb5752-d0dc-4e88-aa14-b1ad0a3a7693",
 CellID->1203546909],
Cell[53091, 1383, 500, 14, 32, "DisplayFormula",ExpressionUUID->"8a5ebb1c-d5cc-4485-a8d7-1efb34698476",
 CellID->808379129],
Cell[53594, 1399, 239, 5, 32, "DisplayFormula",ExpressionUUID->"c169dbe6-ce67-4c2e-a44c-cc411f5b0b77",
 CellID->1446470181],
Cell[53836, 1406, 313, 8, 36, "Text",ExpressionUUID->"dc5ab057-6391-43e7-bb64-57a3de41dee9",
 CellID->722423335],
Cell[CellGroupData[{
Cell[54174, 1418, 1934, 47, 27, "Input",ExpressionUUID->"016e8c2e-64cd-4d75-b433-a7946758b1cb",
 CellID->2043934854],
Cell[56111, 1467, 2117, 74, 92, "Output",ExpressionUUID->"a6837e00-3b12-41db-9ed1-7ef1a3df480f",
 CellID->1450561172]
}, {2}]],
Cell[58240, 1544, 346, 9, 36, "Text",ExpressionUUID->"d36ae15b-d1a2-4743-8750-ff89c44750d0",
 CellID->1726330803],
Cell[CellGroupData[{
Cell[58611, 1557, 631, 15, 25, "Input",ExpressionUUID->"5082b51c-2bd9-4829-a276-33d3ad2cd471",
 CellID->1568066294],
Cell[59245, 1574, 4999, 123, 248, "Output",ExpressionUUID->"fe2a8d8c-2873-4502-9dbc-ab273cf584c7",
 CellID->826107982]
}, {2}]]
}, Open  ]],
Cell[CellGroupData[{
Cell[64290, 1703, 491, 6, 56, "Subsection",ExpressionUUID->"846307f5-2eaf-4f36-94ae-5e64be2f4627",
 CellID->1158490424],
Cell[64784, 1711, 697, 20, 59, "Text",ExpressionUUID->"b8a8d00d-707d-42ac-bd7a-8ed41622ed3e",
 CellID->1661812466],
Cell[CellGroupData[{
Cell[65506, 1735, 300, 7, 54, "Subsubsection",ExpressionUUID->"4ab62bf0-f6f1-4e76-9109-4914fa512335",
 CellID->1138430537],
Cell[65809, 1744, 507, 15, 59, "Text",ExpressionUUID->"8890c490-5844-4b59-a520-b12cc302b565",
 CellID->2057286827],
Cell[66319, 1761, 568, 18, 37, "Text",ExpressionUUID->"d235a26f-63ea-467b-9cb6-e323cebf8e01",
 CellID->1503273659],
Cell[66890, 1781, 201, 5, 32, "DisplayFormula",ExpressionUUID->"d839ea11-dd78-4497-9127-bb694722936d",
 CellID->979255788],
Cell[67094, 1788, 306, 8, 36, "Text",ExpressionUUID->"4f4e8cf3-784c-4f86-aa98-ddfd72b5055f",
 CellID->1717241097],
Cell[CellGroupData[{
Cell[67425, 1800, 1583, 40, 27, "Input",ExpressionUUID->"9c82d9f0-c9bf-482d-9fad-ffb5ff9aa5af",
 CellID->708577063],
Cell[69011, 1842, 668, 15, 63, "Output",ExpressionUUID->"342fa5cc-7e73-4377-9aa2-12c64437f74b",
 CellID->2121983046]
}, {2}]],
Cell[69691, 1860, 441, 12, 36, "Text",ExpressionUUID->"2deef9ee-6d90-4265-8336-650d58d5561b",
 CellID->827290817],
Cell[CellGroupData[{
Cell[70157, 1876, 577, 14, 25, "Input",ExpressionUUID->"6c9e938c-6214-40bc-ac1e-501f86ae9971",
 CellID->786110471],
Cell[70737, 1892, 4557, 114, 241, "Output",ExpressionUUID->"bad2157e-7657-42a3-b246-90f20e91c803",
 CellID->2008861789]
}, {2}]]
}, Open  ]],
Cell[CellGroupData[{
Cell[75340, 2012, 644, 13, 54, "Subsubsection",ExpressionUUID->"72b5ef1f-2a7d-4104-b43c-c6370dca4210",
 CellID->764736412],
Cell[75987, 2027, 544, 14, 59, "Text",ExpressionUUID->"9b27b8f0-7619-4540-9501-e31508c04b75",
 CellID->235249189],
Cell[76534, 2043, 567, 18, 37, "Text",ExpressionUUID->"36c778f2-2ab8-4729-b524-5d749f4f3957",
 CellID->898568374],
Cell[77104, 2063, 297, 7, 32, "DisplayFormula",ExpressionUUID->"024f0371-9810-40ac-ab49-fd718e90db3a",
 CellID->883391814],
Cell[77404, 2072, 312, 8, 36, "Text",ExpressionUUID->"2fdaaf86-7c01-428e-a36f-d31f95b252ab",
 CellID->537870064],
Cell[CellGroupData[{
Cell[77741, 2084, 1661, 42, 27, "Input",ExpressionUUID->"6adaa962-9701-4172-aceb-a701abd2656f",
 CellID->1223495425],
Cell[79405, 2128, 771, 17, 63, "Output",ExpressionUUID->"546a82b1-232e-4b31-84dc-64b75f62d5ee",
 CellID->865604348]
}, {2}]],
Cell[80188, 2148, 433, 12, 36, "Text",ExpressionUUID->"61412fa3-8a2e-4f5b-9383-2d145bad36e6",
 CellID->16714171],
Cell[CellGroupData[{
Cell[80646, 2164, 583, 14, 25, "Input",ExpressionUUID->"0448da14-1336-4b1e-8b84-4420a80d21bd",
 CellID->1435373755],
Cell[81232, 2180, 4610, 116, 246, "Output",ExpressionUUID->"e138b246-b3f0-41b0-b5d1-86f6ab56968b",
 CellID->226090980]
}, {2}]],
Cell[CellGroupData[{
Cell[85876, 2301, 110, 1, 69, "TechNoteSection",ExpressionUUID->"436f7a98-fb5b-4a7c-8adc-1755eff36ad0",
 CellID->1259673378],
Cell[85989, 2304, 230, 3, 29, "SummaryList",ExpressionUUID->"a7178d9f-5359-4289-b355-f74602787985",
 CellID->606253025],
Cell[86222, 2309, 758, 12, 52, "SummaryList",ExpressionUUID->"987b8e00-84e3-4d07-a3d7-9f0ecaa11b1f",
 CellID->1632169807],
Cell[86983, 2323, 541, 15, 75, "SummaryList",ExpressionUUID->"65134c60-012f-4597-952e-f51dd41378b2",
 CellID->2069323118],
Cell[87527, 2340, 401, 9, 52, "SummaryList",ExpressionUUID->"eb9e406d-b2a4-441d-96be-711fe1d88ca6",
 CellID->615981061]
}, Open  ]]
}, Open  ]]
}, Open  ]]
}, Open  ]],
Cell[87979, 2355, 293, 6, 135, "SmallText",ExpressionUUID->"335b811d-f0d1-4160-8ae7-5c3a9270b7f0",
 CellGroupingRules->"SectionGrouping"]
}
]
*)

(* NotebookSignature WTGnaHFTj0EyoiIePi#dvSNG *)
