The Erdős–Oler 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 $m\geq1$, let $L(m)$ be the smallest side length of a closed equilateral triangle that contains $m$ closed disks of radius one with pairwise disjoint interiors. Touching disks and contact with the triangle's boundary are allowed. The radius-one convention fixes the scale. A triangular number is $T_k=k(k+1)/2$. Take $k\geq2$, so both $T_k$ and $T_k-1$ are positive.

Statement recorded in the supplied source.

Is $L(T_k-1)=L(T_k)$ for every integer $k\geq2$? In other words, can deleting one disk from an optimal triangular-number packing ever allow a smaller containing equilateral triangle? [S2]

Short English statement

Does packing one fewer equal disk than a positive triangular number still require an equilateral triangle of the same minimum size?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200026 (AMR-071-0026). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. L. Graham and B. D. Lubachevsky, Dense Packings of Equal Disks in an Equilateral Triangle: From 22 to 34 and Beyond (author preprint, 2004). Abstract and Section 1, pp. 1–2. https://arxiv.org/pdf/math/0406252v1.