Five squares in arithmetic progression over quadratic fields

TL;DR

This paper investigates the existence of five-term arithmetic progressions of squares over quadratic fields, providing criteria, an algorithm for construction, and a classification of solutions over specific fields.

Contribution

It introduces new criteria for existence, develops an algorithm for finding all such progressions, and classifies solutions over certain quadratic fields.

Findings

Only one non-constant progression over Q(√409) is found.

An algorithm for constructing all progressions over quadratic fields is provided.

Several open problems and conjectures are proposed.

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.