Revista Matemática Iberoamericana

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Volume 29, Issue 4, 2013, pp. 1211–1238
DOI: 10.4171/RMI/754

Published online: 2013-12-15

Five squares in arithmetic progression over quadratic fields

Enrique González-Jiménez[1] and Xavier Xarles[2]

(1) Universidad Autónoma de Madrid, Spain
(2) Universitat Autónoma de Barcelona, Spain

We provide several criteria to show over which quadratic number fields $\mathbb{Q}(\sqrt{D})$ there is a nonconstant arithmetic progression of five squares. This is carried out by translating the problem to the determination of when some genus five curves $C_D$ defined over $\mathbb{Q}$ have rational points, and then by using a Mordell–Weil sieve argument. Using an elliptic curve Chabauty-like method, we prove that, up to equivalence, the only nonconstant arithmetic progression of five squares over $–(\sqrt{409})$ is $7^2$, $13^2$, $17^2$, $409$, $ 23^2$. Furthermore, we provide an algorithm for constructing all the nonconstant arithmetic progressions of five squares over all quadratic fields. Finally, we state several problems and conjectures related to this problem.

Keywords: Arithmetic progressions, squares, quadratic fields, elliptic curve Chabauty method, Mordel–Weil sieve

González-Jiménez Enrique, Xarles Xavier: Five squares in arithmetic progression over quadratic fields. Rev. Mat. Iberoamericana 29 (2013), 1211-1238. doi: 10.4171/RMI/754