Diophantine modular solution

So I have math results that I've had for YEARS and it has occurred to me that maybe some of them might be of interest on this blog. And here's one where I found something so simple it puzzles me if it's not known already.

Consider x² - Dy² = F with all integers. Turns out I found there is a very easy way to solve for x and y modularly.

Given x² - Dy² = F where all variables are non-zero integers, with a non-zero integer N for which a residue m exists where m² = D mod N, and with r, any residue modulo N for which Fr^-1 mod N exists then:

2x = r + Fr^-1 mod N

and

2my = r - Fr^-1 mod N

It is EASY to derive so you may see if you can figure it out. Or you can see it derived here.

That result gives solutions to x² - Dy² = F mod N.

This thing is so simple I find it hard to believe it's a new discovery. So I'm emphasizing that and also still looking for it elsewhere. I think it's simple and cool though, even if it's just another re-discovery.

I do wonder if you could use it with, say, factoring, but haven't noticed anything about which I'm sure. And I think part of me just kind of just wants it to be important, you know?

But then again, I don't know if you could do much with it either. So it's just this thing I have and wonder about once in a while.


James Harris