Five squares in arithmetic progression over quadratic fields

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.