Squares in arithmetic progression over cubic fields

TL;DR

This paper proves that cubic number fields cannot contain five squares in arithmetic progression, extending the understanding of squares in arithmetic progressions beyond quadratic fields.

Contribution

It establishes the non-existence of five-term squares in arithmetic progression within cubic number fields, a new result in number theory.

Findings

No cubic number fields contain five squares in arithmetic progression.

Extends the known limitations of squares in arithmetic progressions to cubic fields.

Builds on previous results in quadratic fields by Xarles.

Abstract

Euler showed that there can be no more than three integer squares in arithmetic progression. In quadratic number fields, Xarles has shown that there can be arithmetic progressions of five squares, but not of six. Here, we prove that there are no cubic number fields which contain five squares in arithmetic progression.