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Statement

Setting (p. 2). For $n$ non-overlapping equilateral triangles of side lengths $a_1,\ldots,a_n$ packed inside an equilateral triangle of side $1$, $f(n)$ is the maximum of $a_1+\cdots+a_n$ over all such packings. Comparing areas and Cauchy-Schwarz give $f(n)\le\sqrt n$, and tiling by $k^2$ congruent triangles gives $f(k^2)=k$ for every positive integer $k$. As in Theorem 1, $\epsilon(k)=f(k^2+1)-k$.

Theorem 2 (p. 2, quoted). "$f(k^2+1)=k$ for all $k$ if and only if $\sum_{k\geqslant1}\epsilon(k)$ converges."

Consequence (p. 3, unlabelled). If $f(k^2+1)=k$ for infinitely many $k$, then $f(k^2+1)=k$ for every $k$; this follows from part 1 of Theorem 1.

Proof pointer

Pp. 2--3. The paper first argues that this $f$ obeys hypothesis (*) of Theorem 1. For $f(k^2+1)\ge k$ it points to Fig. 1, an example captioned $n=2$ with side sum $2$; no other case is treated. For the subdivision inequality it follows Praton's argument for squares: cut the unit triangle into the $b\times b$ triangular grid, replace an $a\times a$ corner subgrid by an optimal packing of $n$ triangles scaled by $a/b$, and count $b^2-a^2+n$ triangles of total side $(b^2-a^2)/b+a f(n)/b\le f(b^2-a^2+n)$. The paper notes (p. 3) that the argument works for any shape tileable by a square number of congruent copies similar to it. Theorem 2 then follows from Theorem 1: the forward direction is trivial, and if some $\epsilon(n)>0$, part 2 gives $\epsilon(k)=\Omega(1/k)$, so the series diverges.

Read depth

Claims checked: the setting, Theorem 2, the argument for (*), the remark on p. 3 and the proof were read clause by clause on the page images of arXiv v1. Nothing here is independently reviewed.

Dependencies

Theorem 1. External input named by the paper: I. Praton, Packing squares in a square, Math. Mag. (2008), 358--361, for the subdivision argument.

Source. Anshul Raj Singh, On a square packing conjecture of Erdős, arXiv:2601.22163 (2026); the edition read is named on the source card.

Bears on

• Problem 106: Theorem 2 is the triangle analogue of the problem, which the paper raises on p. 2 as a question similar to Erdős's conjecture; it says nothing about squares by itself. The square case is the paper's Section 4, recorded on its own page.