On a question of Mordell

TL;DR

This paper improves methods for finding integer solutions to the equation x^3 + y^3 + z^3 = k, successfully discovering new solutions for specific k values and completing a decades-old mathematical search.

Contribution

It introduces enhanced computational techniques and implements them on a large volunteer grid, resolving a long-standing mathematical challenge.

Findings

New solutions for k=3 and k=42 found

Completes the search initiated in 1954

Resolves a challenge posed by Mordell in 1953

Abstract

We make several improvements to methods for finding integer solutions to $x^3+y^3+z^3=k$ for small values of $k$. We implemented these improvements on Charity Engine's global compute grid of 500,000 volunteer PCs and found new representations for several values of $k$, including $k=3$ and $k=42$. This completes the search begun by Miller and Woollett in 1954 and resolves a challenge posed by Mordell in 1953.