# Research for “anti-solutions” to diophantine equations.

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)?

• 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. – John Omielan Aug 8 '19 at 23:39
• 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". – Jesse Aug 8 '19 at 23:48
• @Jesse: you need to ask a more specific question. E.g., what integers lie in the range of the function $(x, y) \mapsto \sqrt{x^2 + y^2}$? – Rob Arthan Aug 8 '19 at 23:51
• The Pythagorean triples was only an example. I am looking for any research, in general, for the "anti-solutions" to Diophantine equations. – Jesse Aug 9 '19 at 0:15
• @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. – Adam Bailey Aug 9 '19 at 11:29

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 $$4^r(8s-1)$$.
• 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 $\mathbb Q_2$. – Piquito Aug 9 '19 at 16:49