%0 Journal Article
%T Five squares in arithmetic progression over quadratic fields
%A Enrique Gonz¨˘lez-Jim¨Śnez
%A Xavier Xarles
%J Mathematics
%D 2009
%I arXiv
%R 10.4171/RMI/754
%X 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.
%U http://arxiv.org/abs/0909.1663v4