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

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

:)

• Are you trying to find all integer solutions? – Peter Woolfitt Jan 21 '15 at 21:13
• Welcome to MathSE! You are more likely to get a good answer to your question if you follow a few guidelines. In particular, you need to make your question clear. As @PeterWoolfitt asked, are you looking for all integer solutions, or for just some integer solutions, or are rational or irrational solutions acceptable? – Rory Daulton Jan 21 '15 at 21:21
• Thanks marty, that helps greatly. Yes, I'm only interested in integer values, but 1579 in particular. – Jo pus Jan 21 '15 at 22:46

First of all, reduce 1263465 mod 144, so you then only have to solve $y^2 = a+144x$ where $0 \le a < 144$.
• You can also simplify by $9$. – Yves Daoust Jan 21 '15 at 21:16