Arithmetic progressions of four squares over quadratic fields

TL;DR

This paper investigates the existence of four-term arithmetic progressions of squares over quadratic fields Q(√d), providing partial answers and explicit constructions depending on the value of d.

Contribution

It offers new partial results and explicit examples of four squares in arithmetic progression over quadratic fields, advancing understanding in this area.

Findings

Partial answers depending on d

Explicit constructions of progressions over Q(√d)

Conditions under which such progressions exist

Abstract

Let d be a squarefree integer. Does there exist four squares in arithmetic progression over Q(sqrt{d})? We shall give a partial answer to this question, depending on the value of d. In the affirmative case, we construct explicit arithmetic progressions consisting of four squares over Q(sqrt{d}).