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WindowTitle->13. Diophantine Equations | Introduction to Number Theory
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Cell["\<\
Imagine a problem: how many ways are there to make $2.00 from only nickels \
and quarters?\
\>", "Text",
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Cell[TextData[{
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of nickels and quarter, and all of that would add up to ",
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Cell["\<\
Imagine you have that equation. For now, it does not matter which equation \
exactly. The unknowns in that equation would logically be the number of \
nickels and quarters.\
\>", "Text",
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Cell[TextData[{
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Cell["\<\
So, you put a restraint on the unknowns: the unknowns must be natural \
numbers. Your problem now is to solve that equation in natural numbers.\
\>", "Text",
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Cell["\<\
This is so different from solving equations with no restrictions. As a matter \
of fact, there\[CloseCurlyQuote]s a huge and very important subdivision in \
number theory dedicated to finding solutions to Diophantine equations.\
\>", "Text",
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Cell[TextData[{
 "A ",
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 " is a polynomial equation, usually in two or more unknowns, such that only \
the integer solutions are sought or studied."
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Cell["\<\
Diophantine problems have fewer equations than unknown variables and involve \
finding integers that work correctly for all equations.\
\>", "Text",
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Cell["\<\
Historically, individual Diophantine equations have been studied as puzzles, \
but the development of comprehensive theories to address them is a \
twentieth-century breakthrough.\
\>", "Text",
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Think back to the problem that was already posed: how many ways are there to \
make $2.00 from only nickels and quarters?\
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Mathematically speaking, the problem is: how many integer solutions does this \
equation have?\
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However, this is different from simply solving an equation because:\
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The solutions are restricted because they must be non-negative integers.\
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Cell["This is a classic example of a Diophantine equation.", "Text",
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the integer solutions are studied."
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Cell["Here are some examples of Diophantine equations:", "Text",
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Cell["The typical questions concerning a Diophantine equation are:", "Text",
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Cell["Are there any solutions?", "Item",
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Are there any solutions beyond some that are easily found by inspection?\
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Cell["Can a full list of solutions be computed?", "Item",
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Cell["\<\
These classic problems remained unsolved for hundreds of years until \
mathematicians started appreciating their complexity instead of viewing them \
as mere puzzles.\
\>", "Text",
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Cell["\<\
First, you cannot solve any given Diophantine equation. Moreover, a general \
algorithm for solving all Diophantine equations cannot exist.\
\>", "Text",
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Apart from the simplest cases of Diophantine equations, there isn\
\[CloseCurlyQuote]t a general method for solving them. Outside a certain \
class of equations, each Diophantine equation is treated individually.\
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Now let\[CloseCurlyQuote]s solve a simple Diophantine equation with no \
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There is a general algorithm by which you can solve such equations, but you \
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