Five squares in arithmetic progression over quadratic fields

Abstract

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 \mathbb Q(\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.