Perfect powers that are sums of consecutive cubes

TL;DR

This paper generalizes previous work by fully characterizing all perfect powers that can be expressed as sums of up to 50 consecutive cubes, using advanced number-theoretic techniques.

Contribution

It extends Stroeker's results to all perfect powers, employing descent, linear forms in logarithms, and Frey-Hellegouarch curves.

Findings

All perfect powers as sums of up to 50 consecutive cubes are determined.

The methods successfully handle the complexity of the problem.

The results include new classifications of such perfect powers.

Abstract

Euler noted the relation $6^3=3^3+4^3+5^3$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular Stroeker determined all squares that can be written as a sum of at most $50$ consecutive cubes. We generalize Stroeker's work by determining all perfect powers that are sums of at most $50$ consecutive cubes. Our methods include descent, linear forms in two logarithms, and Frey-Hellegouarch curves.