I have a Diophantine equation $$-900x^2 + y^2 + 1842300 = 0$$ and I would like to find if it has solutions (not the actual solutions, just check if a solution exists). I was told that all I need to do is check the equation modulo the prime divisors of $900$. The prime divisors are $2^2$, $3^2$, and $5^2$ but I'm not real sure how to "check" it.
It looks like $$y^2 + 1842300 \equiv 0 \mod 900$$ so do I start with $2$ and plug in $0$ and $1$ to see if I get $0$ (mod $2$), then go on to $3$ (trying $0$, $1$, and $2$) and then on to $5$ if it is not?
I'm trying to write a program that will determine if $-900x^2 + y^2 + a = 0$ has solutions for a given $a$.