Which integers can be written as the sum of three perfect cubes

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\mathcal C=\{x^3+y^3+z^3:x,y,z\in\mathbb Z\}$. Zero and negative cube summands are permitted. Since an integer cube is congruent to $0$, $1$ or $-1$ modulo $9$, no integer congruent to $4$ or $5$ modulo $9$ lies in $\mathcal C$.

Statement recorded in the supplied source.

Determine $\mathcal C$. In particular, does every integer $N\not\equiv4,5\pmod9$ admit $N=x^3+y^3+z^3$ with $x,y,z\in\mathbb Z$, so that the displayed congruence obstruction is the complete characterization? [S2]

Short English statement

Which integers are sums of three integer cubes, allowing negative cubes and zero?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400111 (AMR-093-0111). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 N. D. Bagis, On the numbers that are sums of three cubes, arXiv:2009.11972v1 (2020) Introduction, equation (1) and Heath--Brown conjecture, p. 1. https://arxiv.org/pdf/2009.11972.
• S3 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Which integers can be written as the sum of three perfect cubes. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.