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Claim. For the pair $(p,q)=(5,2)$, written $(2,5)$ by the authors, the set $\mathcal{R}$ of integers that are sums of numbers $2^a5^b$ no one of which divides another has positive lower density: its counting function satisfies $\mathcal{R}(x)\gg x$. The manuscript On a problem on $d$-complete sequences by Yuchen Ding, Huixi Li, Honghu Liu and Zihan Zhang was posted on ResearchGate and claimed on the site's proof-claims tab on 2026-08-24 as a partial proof, with the AI system GPT 5.6 Sol named as used. Yu and Chen [YuCh22] had shown that the representable numbers have density zero for $q>3$, for $q=3$ with $p>6$ and for $q=2$ with $p>10$; the pair $(5,2)$ lies outside those ranges, and the claim shows that its representable numbers are a positive proportion rather than a null set. The description below follows the claim's summary and notes on the site's proof-claims tab, not the manuscript.

Submission note. Posted to erdosproblems.com as a proof claim by Yuchen Ding, Huixi Li, Honghu Liu, Zihan Zhang (account Honghu_Liu) on 24 August 2026, giving "GPT 5.6 Sol" as the AI used:

If $\{p,q\}\ne\{2,3\}$, Erdős and Lewin asked what can be said about the density of the nonrepresentable integers. Yu and Chen made some progress as described above. We show that for $(p,q)=(2,5)$, the representable integers have positive lower density. Let $\mathcal R$ denote the set of representable integers for $2$ and $5$, and let $\mathcal R(x)$ be its counting function. Set $P=2^{12}$. For every positive integer $N$, we define a suitable class of admissible words

$$> \mathbf r=(r_0,\ldots,r_{N-1})\in\{0,1,\ldots,P-1\}^N. >$$

We construct an injective map $\Phi$ from the admissible words to $\mathcal R$ such that $\Phi(\mathbf r)\leq 10546\,P^{N}$. Using the cycle lemma and a double counting argument, we show that the number of admissible words is at least $57\,P^{N-1}$. Hence $\mathcal R(10546\,P^{N})\gg P^{N},$ which implies the required conclusion. Notes: The key finite input is a family of sets $C(r)=\{(\alpha_{r,i},\gamma_{r,i}):1\leq i\leq t_r\}$, $0\leq r<P$, with $C(0)=\varnothing$, such that

$$> \sum_{(\alpha,\gamma)\in C(r)}2^\alpha(5^\gamma)^{-1}\equiv r\pmod P, >$$

where

$$> 0\leq \alpha_{r,1}<\alpha_{r,2}<\ldots<\alpha_{r,t_{r}}\leq 11\quad\text{and}\quad 0\leq \gamma_{r,1}<\gamma_{r,2}<\ldots<\gamma_{r,t_{r}}\leq 7. >$$

Moreover, if $H(0)=0$ and

$$> H(r)=1+\max\{\gamma:(\alpha,\gamma)\in C(r)\} \qquad(r\ne0), >$$

then the average value of $H(r)$ over $0\leq r<P$ is less than $5$.

The argument. Put $P=2^{12}$. For each positive integer $N$ a class of admissible words $\mathbf{r}=(r_0,\ldots,r_{N-1})$ with entries in $\{0,\ldots,P-1\}$ is defined, and an injective map $\Phi$ sends each admissible word to a representable integer at most $10546\,P^N$. The finite input is a table of sets $C(r)$ of exponent pairs $(\alpha,\gamma)$, one for each residue $r$ modulo $P$, with $C(0)$ empty, strictly increasing coordinates, $\alpha\leq 11$, $\gamma\leq 7$, and $\sum_{(\alpha,\gamma)\in C(r)}2^\alpha 5^{-\gamma}\equiv r\pmod P$, such that the height $H(r)=1+\max\gamma$ (with $H(0)=0$) averages less than $5$ over the residues. A cycle lemma and double counting give at least $57\,P^{N-1}$ admissible words, so $\mathcal{R}(10546\,P^N)\gg P^N$, which is the positive lower density.

Covers. The first question of Problem 1110, the density of the non-representable numbers, for the single pair $(5,2)$: the non-representable numbers do not have density one there, since the representable ones have positive lower density. No other pair, no natural density, and nothing about the second question (coprime non-representable numbers) is claimed. The claim value is proved: the result proves a lower bound for this pair, positive lower density of the representable integers, without determining the density.

Standing. Claimed. The claim is a manuscript on a preprint-sharing site with no journal record found; the site labels the problem open (page last edited 1 April 2026); no review is recorded. The later claim Bécart 2026 describes itself as an extension of this finite-certificate method to the pair $(7,2)$. Nothing on this page is independently reviewed by this project.