Five squares in arithmetic progression over quadratic fields

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.

To appear in Revista Matemática Iberoamericana

Cited by in corpus (3)

• Covering techniques and rational points on some genus 5 curves
• On a conjecture of Rudin on squares in Arithmetic Progressions
• Squares in arithmetic progression over certain non-primitive quartic number fields