AN OBSERVATION CONCERNING THE REPRESENTATION OF POSITIVE INTEGERS AS A SUM OF THREE CUBES

Abstract

In recent years there has been significant progress on the problem of representing integers as a sum of three cubes. Most significantly are the relatively new solutions found by Booker and Sutherland involving integers with as many as 21 digits. In this paper, we look closely at the solutions that have been found, associate to them a pair $(\gamma ,s)$, where $\gamma$ is an algebraic number in the associated pure cubic field, and $s$ is a positive integer, and describe an algorithm to compute this pair, which in some cases appears to be competitive with existing methods to solve the original representation problem.