Sums of Cubes in Quaternion Rings

TL;DR

This paper explores the problem of representing quaternions with integer coefficients as sums of cubes within quaternion rings, establishing bounds on the number of cubes needed for such representations.

Contribution

It introduces bounds for the number of cubes required to represent all integer-coefficient quaternions, extending Waring's Problem to quaternion rings.

Findings

Established global bounds for sums of cubes in quaternion rings

Extended classical Waring's Problem to non-commutative algebraic structures

Provided new insights into quaternion number representations

Abstract

We investigate a version of Waring's Problem over quaternion rings, focusing on cubes in quaternion rings with integer coefficients. We determine the global upper and lower bounds for the number of cubes necessary to represent all such quaternions.