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.