Research for "anti-solutions" to diophantine equations.

Question

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)?

Answer 1

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.