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?



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    $\begingroup$ Are you trying to find all integer solutions? $\endgroup$ – Peter Woolfitt Jan 21 '15 at 21:13
  • $\begingroup$ 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? $\endgroup$ – Rory Daulton Jan 21 '15 at 21:21
  • $\begingroup$ Thanks marty, that helps greatly. Yes, I'm only interested in integer values, but 1579 in particular. $\endgroup$ – 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 $0 \le a < 144$.

Then look up "quadratic residues".

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    $\begingroup$ You can also simplify by $9$. $\endgroup$ – Yves Daoust Jan 21 '15 at 21:16
  • $\begingroup$ I did not do any computation. Lazy me. But this is a good point, in that you want a and 144 (or whatever they reduce to) to be relatively prime. $\endgroup$ – marty cohen Jan 21 '15 at 21:20

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