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

Filip Najman

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

Number Theory 2011-11-24 v1

Authors: Filip Najman

Abstract

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.

Keywords

Diophantine equation Gaussian integers elliptic curves quartic equation Hilbert's result

Cite

@article{arxiv.1005.1528,
  title  = {The Diophantine equation $x^4\pm y^4=iz^2$ in Gaussian integers},
  author = {Filip Najman},
  journal= {arXiv preprint arXiv:1005.1528},
  year   = {2011}
}

Comments

5 pages, to appear in Amer. Math. Monthly

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