Solving $y^2 = 1263465 + 144x$ for integers $x,y$

Question

I've thrown this equation up as part of some research I'm doing.

$$y^2 = 1263465 + 144x$$

I was hoping there is a quick way to solve this without stepping through all the values. The value I'm interested in is $x = 1579$, but other $x$ values satisfy for $y$ as well, although I'm not bothered about those. Am I right in thinking that factorising $1263465$ might simplify the problem somewhat?

Cheers

:)

Answer 1

First of all, reduce 1263465 mod 144, so you then only have to solve $y^2 = a+144x$ where $0 \le a < 144$.

Then look up "quadratic residues".