Arithmetic progressions of squares, cubes and $n$-th powers

TL;DR

This paper investigates the maximum length of primitive arithmetic progressions composed of squares, cubes, and higher powers, providing sharp upper bounds for such sequences.

Contribution

It offers new sharp upper bounds on the length of primitive arithmetic progressions of various powers, advancing understanding of their structure.

Findings

Sharp upper bounds established for progressions of squares and cubes

Extended bounds for higher powers in arithmetic progressions

Enhanced understanding of the limitations of such progressions

Abstract

In this paper we continue the investigations about unlike powers in arithmetic progression. We provide sharp upper bounds for the length of primitive non-constant arithmetic progressions consisting of squares/cubes and $n$-th powers.