Representing integers as a sum of three cubes

TL;DR

This paper advances methods for expressing integers as sums of three cubes, overcoming previous barriers and demonstrating improved computational efficiency, with ongoing hybrid approaches for unresolved cases.

Contribution

It develops new techniques for representing integers as sums of three cubes and overcomes key barriers in the existing methods.

Findings

Method shows favorable time estimates compared to previous approaches.

Recent computations confirm the efficiency of the new method.

Hybrid approaches are being developed for remaining unsolved cases.

Abstract

In this article we further develop methods for representing integers as a sum of three cubes. In particular, a barrier to solving the case $k=3$, which was outlined in a previous paper of the second author, is overcome. A very recent computation indicates that the method is quite favourable to other methods in terms of time estimates. A hybrid of the method presented here and those in a previous paper is currently underway for unsolved cases.