An observation concerning the representation of positive integers as a sum of three cubes (Q6158201)

The diophantine equation \(x^3+y^3+z^3=k\) has been a subject of interest since \textit{L. J. Mordell} [J. Lond. Math. Soc. 28, 500--510 (1953; Zbl 0051.27802)] asked about its solutions for \(k=3\), other than the two obvious ones. Only recently in [Proc. Natl. Acad. Sci. USA 118, No. 11, Article ID e2022377118, 11 p. (2021; \url{doi:10.1073/pnas.2022377118})], \textit{A. R. Booker} and \textit{A. V. Sutherland} found a third (large) solution to this equation. The author here presents an algorithm to solve this equation using the arithmetic of cubic fields. It is assumed that \(x-y=t\) is small in \(x^3-y^3=z^3+k\) (rewriting the equation). Then for fixed values of \(t\), the problem is reduced to looking for cube roots of \(-k\) modulo \(t\). The values of \(t\) (or a multiple) are chosen as norms of elements in the ring of integers of the cubic field \(\mathbb{Q}(k^{1/3})\).