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Statement

Setting (p. 1 and p. 3). $f(n)$ is the maximum of $a_1+\cdots+a_n$ over the side lengths $a_1,\ldots,a_n$ of $n$ non-overlapping squares packed inside a unit square; Cauchy-Schwarz gives $f(n^2)=n$, and Erdős conjectured $f(n^2+1)=n$ for all positive integers $n$. As in Theorem 1, $\epsilon(k)=f(k^2+1)-k$.

Section 4 (p. 3, unlabelled). The paper asserts that this $f$ satisfies hypothesis (*) of Theorem 1, "by the same argument as in the case of the equilateral triangle", and concludes:

1. Erdős's conjecture holds if and only if $\sum_{k\ge1}\epsilon(k)$ converges.
2. If $f(k^2+1)=k$ for infinitely many $k$, the conjecture is true.

Section 5 (p. 3, unlabelled). For a parallelogram with sides $1$ and $x$ covered by similar parallelograms with sides $a_i$ and $a_i x$ ($i=1,\ldots,n$), $f(n)$ is redefined as the maximum of $a_1+\cdots+a_n$; the paper asserts that this $f$ satisfies (*), since such a parallelogram tiles by a square number of congruent similar copies, and that the results of Section 4 hold for it as well.

Proof pointer

P. 3. No separate proof is printed. The claim that the square $f$ obeys (*) refers back to the triangle argument on pp. 2--3 (the subdivision of Theorem 2's proof, after Praton), and the conclusions then follow from Theorem 1 as Theorem 2 does. The paper does not write out the bound $f(m^2+1)\ge m$ for squares.

Read depth

Claims checked: Sections 4 and 5 were read clause by clause on the page images of arXiv v1. Nothing here is independently reviewed.

Dependencies

Theorem 1 and the subdivision argument given for Theorem 2.

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: Section 4 reformulates the problem's conjecture as the convergence of $\sum_{k\ge1}(f(k^2+1)-k)$ and reduces it to its truth for infinitely many $k$. It proves neither the conjecture nor its negation.