Five squares in arithmetic progression over quadratic fields

We provide several criteria to show over which quadratic number fields Q(root 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 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(root 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.