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

preprint2010arXivOpen access

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

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The Diophantine equation x4±y4=iz2x^4\pm y^4=iz^2x4±y4=iz2 in Gaussian integers

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