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arXiv · 0909.1663

Five squares in arithmetic progression over quadratic fields

Enrique González-Jiménez·Xavier Xarles
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Abstract

We give several criteria to show over which quadratic number fields Q(sqrt{D}) there should exists a non-constant arithmetic progressions of five squares. This is done by translating the problem to determining when some genus five curves C_D defined over Q have rational points, and then using a Mordell-Weil sieve argument among others. Using a elliptic Chabauty-like method, we prove that the only non-constant arithmetic progressions of five squares over Q(sqrt{409}), up to equivalence, is 7^2, 13^2, 17^2, 409, 23^2. Furthermore, we give an algorithm that allow to construct all the non-constant arithmetic progressions of five squares over all quadratic fields. Finally, we state several problems and conjectures related to this problem.

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Enrique González-Jiménez, Xavier Xarles. 2013-01-24. Five squares in arithmetic progression over quadratic fields. https://doi.org/10.4171/rmi%2F754

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