Find the representation of solutions of $x^2+5y^2=z^2$ when $\gcd(x,y）=1$

Asked Dec 15 '22 at 05:41

Active Dec 15 '22 at 10:05

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I want to show that for $x^2+5y^2=z^2$ and $\gcd(x,y）=1$, there are 2 cases:

$x$ is odd and $y$ is even:

then $x=\pm(r^2+5s^2),~y=2rs,~z=r^2+5s^2$
$x$ is even and $y$ is odd:

then $x=\pm(2r^2+2rs-2s^2),~y=2rs+s^2,~z=2r^2+2rs+3s^2$

Any hints will be appreciated.

edited Dec 15 '22 at 10:05

Gary

asked Dec 15 '22 at 05:41

Gang men

2

Does this answer your question? The solutions to $x^2+5=y^2$. - found using an Approach0 search. Note the second part of this other question asks about solving it over rationals, with the accepted answer explaining it's equivalent to solving $x^2+5z^2=y^2$ over co-prime integers, and explains how to do that. – John Omielan Dec 15 '22 at 06:21
@JohnOmielan Thank you so much but maybe I need more information to complete the problem – Gang men Dec 15 '22 at 06:29
1

Gang men, you can search this site next time, since many people have written very detailed solutions for such Diophantine equations. – Dietrich Burde Dec 15 '22 at 15:30
@DietrichBurde Thank you for you knidness and I have collected the site – Gang men Dec 16 '22 at 09:59

0 Answers0