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.
A quick spot of googling found Two fourths and a square.
Look at pattern six: the unique coprime solution is $1^4+1^4=2 \cdot 1^2 .$