Proving that an equation doesn't have integer solutions I need to prove that there are no integer solutions for a bunch of equations like the following:
$$15x^2 - 7y^2 = 9$$
I was able to solve some simpler ones by picking a dividend and looking into it's remainder table. But it's not working for the others. How should I start thinking this kind of problem? It's from my algebra class and we are looking into divisibility and congruence. Thanks!
Hint: Maybe note first that $3$ must divide $y$. Let $y=3t$. Then we are looking at $15x^2-(7)(9t^2)=9$. So $3$ must divide $x$. Let $x=3s$. We end up with
$$15s^2-7t^2=1.$$
Now work modulo $3$.
Call $x^2=a$ and $y^2=b$ then we have $15a-7b=9$ which is a diaphante equation. It's solution can be found by Euclid's algorithm, but also there is a theorem that no solutions exist iff $9$ isn't a multiple of the $gcd$ of $15$ and $7$. 

In this case there are solutions, so you can find them, then show if they can be or can't be 2 squares.