I'm interested in finding if an integer solution exists to the equation $ax^2 + bx + cy + d = 0$
I found Dario Alpern's website https://www.alpertron.com.ar/QUAD.HTM which has a solver that seems to determine if a solution exists by trying to solve an equation of that form using module 9, 16, and 25. For example, $x^2 + 5 x + 15 y + 50 = 0$ gives "No solutions found using mod 9, so there are no integer solutions." but $x^2 + 15 x + 5 y + 50 = 0$ gives "There are solutions, so we must continue.".
My question is: how does module 9, 16, and 25 allow us to know if a solution exists and where can I find more information? Internet search didn't lead me anywhere - probably because I'm not using the right keywords.
To be clear, I am not interested in actually finding solutions; only whether or not a solution exists.
$x^2+5x+15y+50\equiv x^2+14x+15y+50=(x+7)^2+1+15y\bmod9$. Now $15y$ is a multiple of 3, but $u^2+1$ isn't, so there is no solution modulo 3, never mind modulo 9.
In general, for your type of equation, $ax^2+dx+ey+f=0$, it makes sense to work modulo $e$: $ax^2+dx+f\equiv0\bmod e$. Completing the square on the left side, you will wind up with a congruence of the form $z^2\equiv g\bmod e$, so all you have to do is work out whether $g$ is a quadratic residue (that is, a "square") modulo $e$, and any intro Number Theory text will tell you how to do that, using quadratic reciprocity.