---

Claim. There is a set $A$ of positive integers such that every sufficiently large integer is $n^2+a$ with $n\ge0$ and $a\in A$, and

$$\limsup_{x\to\infty}\frac{A(x)}{\sqrt x}\le\frac{13\sqrt2}{4}\approx4.596,$$

where $A(x)$ counts the elements of $A$ up to $x$. Hence the smallest possible value of the limsup in the first question of Problem 33 is at most $13\sqrt2/4$. The write-up, an untitled and unsigned three-page PDF posted with its LaTeX source in a public GitHub repository (the preprint link, pinned to its upload commit of 7 September 2026), builds $A$ scale by scale. For $q=3^k$ and $L_k=(q^2+3)/4$ it takes from $[L_k,L_{k+1})$ the integers congruent to $0$, $1$ or $2$ modulo $q$ below $q^2$, those congruent to $0$, $1$, $2$, $q$, $q+1$ or $q+2$ modulo $3q$ in $[q^2,2q^2)$, and those congruent to $0$, $1$ or $2$ modulo $3q$ from $2q^2$ on, together with finitely many small integers. Coverage rests on square roots modulo $q$ of the residues congruent to $1$ modulo $3$, lifted by Hensel's lemma. A large $m$ below $q^2$ is reached through one root $r$; above $q^2$ the write-up uses two or three of the roots $r$, $q-r$ and $q+r$, whose remainders $m-r^2$, $m-(q-r)^2$ and $m-(q+r)^2$ lie in different classes modulo $3q$. Counting with $x=yq^2$ gives $A(x)/\sqrt x=2\sqrt y+5/(2\sqrt y)+o(1)$ for $1\le y\le2$, which is largest at $y=2$, where it equals $13/(2\sqrt2)$; the ranges $1/4\le y\le1$ and $2\le y\le9/4$ give at most this.

Submission note. Posted to the erdosproblems.com forum by donteatllamas (account llamaboy) on 7 September 2026, in the problem's thread:

With the help of GPT, I think I can improve the $\limsup \frac{\lvert A\cap{1,\ldots,N}\rvert}{N^{1/2}}$ bound to $\frac{13\sqrt{2}}{4}$. The proof is formalized here.

Covers. An upper bound of $13\sqrt2/4\approx4.596$ for the smallest possible limsup, the first question of Problem 33. The claim does not determine the smallest limsup and asserts no optimality of its set. The liminf question is not covered.

Claimant and assistance. The write-up prints no author's name. The forum user donteatllamas (account llamaboy) posted it, saying that it was made with the help of GPT. The repository belongs to the GitHub account LeanMeanBean, which gives no name, and was created on 7 September 2026. No real name is known, so the page lists no author. The post calls the proof formalized, but the repository holds only the PDF and its LaTeX source; no Lean file is posted.

Relation to the other upper bounds. The bound lies below the $2\varphi^{5/2}\approx6.66$ of van Doorn's construction, the bound the erdosproblems.com remarks record. Plasma AI's pending claim gives $9/2$, below this bound.

Depends on. No page of this wiki.

Standing. Claimed. The thread held no review of the write-up when erdosproblems.com froze its comments on 6 October 2026, and the erdosproblems.com remarks, last edited 27 December 2025, predate it; the problem's proof-claims tab holds no claim (accessed 2026-10-18). There is no refereed publication and no Lean proof, and this corpus has not checked the argument.