I have done some searches on the internet to find any studies that deal with the non-solutions of Diophantine equations. I'm asking for any research articles or web links you know of that deal with the "Anti"-solutions of Diophantine equations. For example: say there is a Diophantine equation z=f(x,y) then find some solution that describes all of z that does not satisfy f(x,y) for the integers or whole number. A particular example would be the Pythagorean triples. What would be the values of c which do not satisfy sqrt(a^2+b^2)?
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3$\begingroup$ Wouldn't the "anti-solutions" just be the set of values in the set you're dealing with (e.g., integers) which aren't solutions? Thus, studying one is really implicitly studying the other one as well. $\endgroup$– John OmielanAug 8, 2019 at 23:39
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2$\begingroup$ I believe that part of what I want to know is whether the set is infinite or finite. Sometimes it is 'easy' to show that there are an infinite number of solutions, but can be possibly impossible to show that there are an infinite number of "anti-solutions". $\endgroup$– JesseAug 8, 2019 at 23:48
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$\begingroup$ @Jesse: you need to ask a more specific question. E.g., what integers lie in the range of the function $(x, y) \mapsto \sqrt{x^2 + y^2}$? $\endgroup$– Rob ArthanAug 8, 2019 at 23:51
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$\begingroup$ The Pythagorean triples was only an example. I am looking for any research, in general, for the "anti-solutions" to Diophantine equations. $\endgroup$– JesseAug 9, 2019 at 0:15
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1$\begingroup$ @Jesse An example that may illustrate your comment. It is easy to show that there are an infinite number of values of $z$ for which $w^3+x^3+y^3=z^3$ has a solution in positive integers. But so far as I am aware it is not known whether the number of values of $z$ for which there is no solution is finite or infinite. $\endgroup$– Adam BaileyAug 9, 2019 at 11:29
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COMMENT.- Each time a characterization is established for an integer to satisfy an equation, the "anti-solutions" of the corresponding diophantine equation are automatically implicitly established. For example, we have the well-known Lagrange's three square theorem (also proved by Gauss) that characterizes integers that are the sum of three squares:
Theorem.- An integer $n$ is a sum of three squares if and only if it is not of the form $4^r(8s-1)$ where $r,s\in\mathbb Z$.
"Corollary".- The set of all the anti-solutions of the diophantine equation $$w=x^2+y^2+z^2$$ is formed for all the integers of the form $4^r(8s-1)$.
Personally speaking I do not see significant relevance to that result.
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$\begingroup$ Do you know of any research that deals with how you would find such an equation? How do you do the transformation of w=x^2+y^2+z^2 --> 4^r(8s-1) $\endgroup$– JesseAug 9, 2019 at 0:49
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3$\begingroup$ It is a "strong" result of Gauss and is not easy to establish. The great Serre prove it in his book "Cours d'Arithmétique" like this: (1) WLOG assumes that $n$ is non-null and then, by virtue of a result in the $2$-adic field $\mathbb Q_2 $ the condition $n = 4 ^ r (8s-1)$ is equivalent to that $-n$ is a square in $\mathbb Q_2 $. (2) States that for the rational non-null $a$ it is representable on $\mathbb Q$ by the quadratic form $x_1^2 + x_2^2+ x_3^2$ it is necessary and sufficient that $a\gt0$ and that $-a$ is not a square in $\mathbb Q_2 $. $\endgroup$– PiquitoAug 9, 2019 at 16:49