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I would like to know any way of solving the diophantine equation $x^4+y^4=2z^2$. Or ideas that seem worth trying out.

By solving I mean fining all solutions and proving there are no more.

Keith Conrad showed how to reduce this equation to a different one which was solved by Fermat in his notes about Fermat descent, other than I have no ideas how to solve it. I tried to do descent on it directly but that seems completely impossible so I am interested in other techniques. Thanks very much.

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    page 451 of *Number Theory: Analytic and modern tools* by Henri Cohen shows how to reduce it to an elliptic curve.2011-03-12

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A quick spot of googling found Two fourths and a square.

Look at pattern six: the unique coprime solution is $1^4+1^4=2 \cdot 1^2 .$

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    @Henry: Yes, I meant coprime solution, though I should have stated it explicitly.2011-03-12