Pollock's tetrahedral-number conjecture

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.

For $n\in\mathbb Z_{\ge0}$ define the tetrahedral number $T_n=n(n+1)(n+2)/6$. The term $T_0=0$ permits a representation with fewer than five terms to be padded to five; repetitions are permitted.

Statement recorded in the supplied source.

For every $N\in\mathbb N$, do there exist $n_1,\ldots,n_5\in\mathbb Z_{\ge0}$ such that $N=T_{n_1}+\cdots+T_{n_5}$? [S2]

Short English statement

Can every positive integer be written as a sum of at most five tetrahedral numbers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400041 (AMR-093-0041). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Zarathustra Brady, Sums of seven octahedral numbers, arXiv:1509.04316v2 (2015) Introduction, p. 1, Pollock conjectures. https://arxiv.org/pdf/1509.04316.
• S3 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Pollock's tetrahedral-number conjecture. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.