I would like to derive all the parametrizations for the nontrivial solutions of this Diophantine equation:
$ x^4+y^4+z^4=9t^2 $
I already know that with the Fauquembergue's parametrization I can find infinite solutions (essentially multiplying by 3 the terms in the parametrization, or in the Pythagorean triple), like this one:
$ 60^4 + 45^4 + 36^4 = 9(1443)^2 $
$ 108^4+135^4+180^4 = 9(12987)^2 $
But I even know that there are more solutions (I found the last two from $ x^4+y^4+z^4=t^4 $):
$(155, 260, 296, 37747)$
$(95800, 217519, 414560, 59496731787)$
$ (2682440, 15365639, 18796760, 141668657747643) $
and they don't seem to come out from the result I found on here (none of the fourth powers is divisible by 3).
So which kind of parametrization will give those solutions?