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WindowTitle->20. Diophantine Equations: Hilbert\:2019s Tenth Problem | \
Introduction to Number Theory
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Hilbert pointed out the most important unsolved mathematical problems that \
the twentieth century was to inherit from the nineteenth century.\
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At the time, those problems were unsolved, and many of them were highly \
influential for twentieth-century mathematics.\
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At the Paris conference, Hilbert presented 10 of the problems. The complete \
list of 23 problems was published after.\
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Mathematicians have found solutions of many Diophantine equations and have \
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Unfortunately, for different types of Diophantine equations, and even for \
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\
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Given a Diophantine equation with any number of unknown quantities and with \
rational integral numerical coefficients: To devise a process according to \
which it can be determined in a finite number of operations whether the \
equation is solvable in rational integers.\
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The words \[OpenCurlyDoubleQuote]process\[CloseCurlyDoubleQuote] and \
\[OpenCurlyDoubleQuote]finite number of operations\[CloseCurlyDoubleQuote] \
mean Hilbert was asking for an algorithm. \
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So Hilbert was asking for a general algorithm to decide whether a given \
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integers.\
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It\[CloseCurlyQuote]s worthy to note that Hilbert\[CloseCurlyQuote]s problem \
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It\[CloseCurlyQuote]s a decision problem, a problem consisting of infinitely \
many individual problems, each of which requires a \[OpenCurlyDoubleQuote]yes\
\[CloseCurlyDoubleQuote] or \[OpenCurlyDoubleQuote]no\[CloseCurlyDoubleQuote] \
answer.\
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Hilbert asked about solvability in integers. You can also consider the \
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It turns out that any general algorithm that can decide whether a given \
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Every natural number is the sum of the squares of four integers (Lagrange \
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You simply replace every parameter with the sum of squares of four extra \
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Every integer can be written as the difference of two natural numbers. \
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You can replace every parameter that ranges in integers with the difference \
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The work on the problem has been in terms of solutions in natural numbers, as \
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integers.\
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Cell["\<\
Hilbert\[CloseCurlyQuote]s tenth problem has been solved, and it has a \
negative answer.\
\>", "Text",
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Cell["\<\
There is no algorithm that, for a given arbitrary Diophantine equation, would \
tell whether or not the equation has a solution.\
\>", "Text",
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Cell["So, Hilbert\[CloseCurlyQuote]s tenth problem is an undecidable \
problem.", "Text",
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Cell[TextData[{
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Cell["Hilbert\[CloseCurlyQuote]s Tenth Problem in a Broader Sense", \
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Cell["\<\
So the original statement of the tenth problem has a negative solution. What \
about solving Diophantine equations in rational numbers? \
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Cell[TextData[{
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is the same as solving ",
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Cell[TextData[{
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Such equations represent only a subclass within the set of all Diophantine \
equations, and it is possible that there is a corresponding algorithm for \
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\>", "Text",
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Cell["Thus, you can understand the tenth problem in two senses:", "Text",
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Cell["\<\
Narrower sense: the decision problem of determining the solvability of an \
arbitrary Diophantine equation in integers (literally as the problem was \
stated)\
\>", "Item",
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Cell["\<\
Broader sense: the decision problem of determining the solvability of an \
arbitrary Diophantine equation in rational numbers\
\>", "Item",
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Cell["\<\
In the narrower sense, the tenth problem is closed; however, it is still open \
in the broader sense.\
\>", "Text",
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Cell["\<\
Solving equations in rational numbers remains one of the most important open \
cases of Hilbert\[CloseCurlyQuote]s tenth problem taken in a broader sense. \
\>", "Text",
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Cell["Summary", "TechNoteSection",
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Cell[TextData[{
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Cell[TextData[{
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equations."
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Cell["\<\
Hilbert\[CloseCurlyQuote]s problem is not concerned with finding the \
solutions. It only asks if you can determine, in general, whether solutions \
exist.\
\>", "SummaryList",
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Cell["\<\
Hilbert\[CloseCurlyQuote]s tenth problem has been solved, and it has a \
negative answer: no such algorithm exists.\
\>", "SummaryList",
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Cell[TextData[{
 "The impossibility of obtaining a general solution was the result of the \
combined work of ",
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 ", ",
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