Does there exists a nontrivial positive integer solution with $x\ne y,$ of $$x^4+y^4=2z^2.$$


marked as duplicate by Dietrich Burde, M Turgeon, Jyrki Lahtonen, Rick Decker, Fly by Night Aug 3 '14 at 18:26

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  • $\begingroup$ See math.stackexchange.com/questions/26591/…. $\endgroup$ – Dietrich Burde Aug 3 '14 at 16:55
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    $\begingroup$ The answer referred to in the previous comment is simply a reference to a possibly evanescent web site. $\endgroup$ – André Nicolas Aug 3 '14 at 16:58
  • $\begingroup$ This may be a better duplicate. For the reason pointed out by André. $\endgroup$ – Jyrki Lahtonen Aug 3 '14 at 18:12
  • $\begingroup$ @AndréNicolas: Sorry, yes. It should have been the other web site. $\endgroup$ – Dietrich Burde Aug 3 '14 at 20:13
  • $\begingroup$ The fact that $x^4\pm y^4=z^2$ has no non-trivial solutions goes back to Fermat, and is standard first number theory course material. A solution that uses this seems perfectly reasonable. $\endgroup$ – André Nicolas Aug 4 '14 at 4:20

First, I suppose that you know there no exist $x,y,z\in\Bbb Z^+$ such that $$x^4-y^4=z^2.$$ You might have seen this while proving FLT for $n=4$ and further theorems. I can give you a hint for this if you need.

If $x^4+y^4=2z^2$, we have $$z^4-(xy)^4=\left({x^4-y^4\over2}\right)^2.$$ Due to the above result, $z=0$ or $x=0$ or $y=0$ or $|x|=|y|$, and these are all trivial solutions.

Thus, there is no non-trivial solution.


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