Squares in arithmetic progression over number fields

TL;DR

This paper establishes an upper bound on the number of perfect squares in arithmetic progressions over number fields, showing the bound is 5 for quadratic fields and extends to higher powers.

Contribution

It proves a degree-dependent upper bound for squares in arithmetic progressions over number fields, generalizing to higher powers beyond squares.

Findings

Maximum of 5 squares in quadratic fields

Bound depends only on the degree of the number field

Generalization to higher powers $k>1$

Abstract

We show that there exists an upper bound for the number of squares in arithmetic progression over a number field that depends only on the degree of the field. We show that this bound is 5 for quadratic fields, and also that the result generalizes to $k$-powers for $k>1$.