JournalsrmiVol. 29, No. 4pp. 1211–1238

Five squares in arithmetic progression over quadratic fields

  • Enrique González-Jiménez

    Universidad Autónoma de Madrid, Spain
  • Xavier Xarles

    Universitat Autónoma de Barcelona, Spain
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Abstract

We provide several criteria to show over which quadratic number fields Q(D)\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 CDC_D defined over Q\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 727^2, 13213^2, 17217^2, 409409, 23223^2. 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.

Cite this article

Enrique González-Jiménez, Xavier Xarles, Five squares in arithmetic progression over quadratic fields. Rev. Mat. Iberoam. 29 (2013), no. 4, pp. 1211–1238

DOI 10.4171/RMI/754