The Diophantine equation $x^4\pm y^4=i{z}^2$ in Gaussian integers

Abstract

In this note we find all the solutions of the Diophantine equation $x^4\pm y^4=i{z}^2$ using elliptic curves over $\mathbb{Q}(i)$. Also, using the same method we give a new proof of Hilbert's result that the equation $x^4\pm y^4=z^2$ has only trivial solutions in Gaussian integers.

arXiv comment5 pages, to appear in Amer. Math. Monthly