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## All solutions to Diophantine Equation $x^2 - 8 y^2 = m^2$ [closed]

I'm looking for solutions to an equation of form $m =\sqrt{x^2 - 8y^2}$. I know that $m$ is a positive integer and so the inside of the square root has to be complete square. So I'm stuck with this diophantine equation $x^2 - 8y^2 = m^2$. In the case where $m = \pm1$ that's Pell Equation which I know how to find solutions to. I also know that I can find infinitely many solutions because solutions to $x^2 - 8y^2 = 1$ are also solutions to $(mx)^2 - 8(my)^2 = m^2$ but that doesn't capture all solutions.

This site seems to derive the solutions by solving $s^2 - 8 = 0$ mod $m^2$ but I don't know why and I don't know what to do with it.

Most papers I find on the topic say they have a solution if $\sqrt{8} > m^2$ but that's not sufficient here.

For example: $m = 7$ -> $x^2 - 8y^2 = 49$ and I want to be able to arrive at $x = 11; y = 3$ and other solutions where $x$ and $y$ aren't $0$ mod $7$.

Any help is appreciated. Thank you.

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This site is intended for questions of mathematical research, so your question is off-topic. Worse, the people at Project Euler have made it abundantly clear that contestants are not to seek help on internet sites. – Gerry Myerson Nov 15 at 12:06
I am trying to better my math knowledge. This is definitely not off-topic. And I obviously am not discussing a solution to a problem. Just asking for math principles for solving a diophantine equation of a specific form. Had I been doing a masters thesis on it, this same very question wouldn't have been closed. Hell, half the questions with diophantine-equation tag are inspired by PE. I don't see how my question is any different. – McTrafik Nov 15 at 18:11
"Had I been doing a masters thesis on it, this same very question wouldn't have been closed." Yes it would. BTW: "off-topic" is a shorthand for a topic not suitable for this site, which is geared towards those working at a professional level. – Todd Trimble Nov 15 at 18:20
So, McTrafik, you got here 12 hours ago, but you're ready to tell those of us who have been here for years what is and what isn't off-topic? I don't think so. – Gerry Myerson Nov 15 at 22:06
@Todd Thank you. Will repost in math.stackoverflow which seems to be a more suitable place. @Gerry If you went somewhere were I've been for years, I would've treated you better. Shame on you. – McTrafik Nov 16 at 9:11