The Diophantine equation $x^4\pm y^4=iz^2$ in Gaussian integers

Details The Diophantine equation $x^4\pm y^4=iz^2$ in Gaussian integers The Diophantine equation $x^4\pm y^4=iz^2$ in Gaussian integers

2010-05-10

arxiv.org/abs/1005.1528v1

Amer. Math. Monthly 117 (2010), 637-641

Mathematics

Number Theory

5 pages, to appear in Amer. Math. Monthly

Scientific paper

In this note we find all the solutions of the Diophantine equation $x^4\pm
y^4=iz^2$ 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.

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