Are solutions of the Diophantine equation $x^4+y^4+z^4=2t^4$ well-known?
I give a solution:
$x=m^2-n^2, y=m^2-2mn, z=n^2-2mn, t=m^2+n^2-mn$
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2$\begingroup$ MathWorld says "there are many solutions" $\endgroup$– Carlo BeenakkerCommented Jun 24, 2017 at 17:38
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6$\begingroup$ This is an example of a K3 surface. You have found a copy of $\mathbb P^1_{\mathbb Q}$ sitting inside the surface. A general conjecture says that there is some number field $K/\mathbb Q$ such that the $K$-rational points on the surface are Zariski dense. If you are interested in questions of this sort, you should read about K3 surfaces, both their geometry and their arithmetic. $\endgroup$– Joe SilvermanCommented Jun 24, 2017 at 18:17
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2$\begingroup$ Possible duplicate of Parametrizing the solutions to a diophantine equation of degree four $\endgroup$– jeqCommented Jun 25, 2017 at 1:25
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