Five squares in arithmetic progression over quadratic fields

Enrique González-Jiménez, Xavier Xarles

Research output: Contribution to journalArticleResearchpeer-review

4 Citations (Scopus)

Abstract

We provide several criteria to show over which quadratic number fields ℚ(√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 CD defined over 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 ℚ(√409) is 72, 132, 172, 409, 232. 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. © European Mathematical Society.
Original languageEnglish
Pages (from-to)1211-1238
JournalRevista Matematica Iberoamericana
Volume29
DOIs
Publication statusPublished - 1 Dec 2013

Keywords

  • Arithmetic progressions
  • Elliptic curve Chabauty method
  • Mordell-Weil sieve
  • Quadratic fields
  • Squares

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