Pollock's octahedral-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 $m\in\mathbb Z_{\ge0}$ define the octahedral number $O_m=(2m^3+m)/3$. These are nonnegative integers, with $O_0=0$. Thus a sum of at most seven positive octahedral numbers can be padded with zeros to a sum of exactly seven members of this sequence.

Statement recorded in the supplied source.

Is it true that for every $n\in\mathbb N$ there exist $m_1,\ldots,m_7\in\mathbb Z_{\ge0}$ such that $n=\sum_{j=1}^7 O_{m_j}$? [S2]

Short English statement

Can every positive integer be written as the sum of at most seven octahedral numbers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400177 (AMR-093-0177). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Zarathustra Brady, Sums of seven octahedral numbers, arXiv:1509.04316v2 (2015), Section 1, Pollock conjecture and definition of octahedral numbers, pp. 1--2. https://arxiv.org/pdf/1509.04316.