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Claim. Write $Y_K(p,q)=\{p^aq^b : a\geq 0,\ 0\leq b\leq K\}$ for coprime integers $p,q>1$, and let $K(p,q)$ denote the smallest $K\geq 0$ for which this set is complete. Then $K(p,q)\leq 2p-3$, and $K(2,q)=0$ by binary expansion. Since $Y_K(p,q)$ is a subset of $\{p^aq^b\}$, the claim implies the statement of Problem 246 and bounds the exponent range that Davenport observed could be bounded; the authors present it as an improvement of a bound $4p-5$ they attribute to Bergelson and Simmons, the bound that Fang and Chen's quantitative form (p. 302) also records as Bergelson and Simmons's theorem of 2017. The manuscript, "A $2p-3$ Upper Bound for a Bounded-Exponent Refinement of Erdos Problem 246", is a Zenodo deposit published on 2026-09-03 (UTC; the record's publication date field reads 2026-09-04) under a Creative Commons Attribution 4.0 license; it is not held and its proof was not reconstructed here.

Submission note. Posted to erdosproblems.com as a proof claim by Zhao Song, Song Yue (account magiclinux) on 4 September 2026, giving "ChatGPT 5.6 Sol" as the AI used:

This submission gives a quantitative strengthening of Erdős Problem 246. For coprime integers p,q > 1, let K(p,q) be the least nonnegative integer K such that the set Y_K(p,q) = {p^a q^b : a ≥ 0 and 0 ≤ b ≤ K} is complete. We prove that K(p,q) ≤ 2p−3. For p = 2, binary expansion gives the stronger result K(2,q) = 0. For p ≥ 3, the proof divides Y_{2p−3}(p,q) into two disjoint blocks of p−1 consecutive q-levels. The finite subset sums of the lower block form a thick set: they contain arbitrarily long intervals of consecutive integers. This follows by extracting a complete residue digit system modulo p from the subset sums of 1,q,...,q^{p−2}, and then applying a one-dimensional self-affine tiling result. Notes: The upper block has finite deficit, so its finite subset sums have bounded gaps. The sum of a thick set and a set with bounded gaps is cofinite. Since the two blocks are disjoint, their subset-sum representations can be combined without repeating an element. This proves that Y_{2p−3}(p,q) is complete.

Argument, in outline. For $p\geq 3$ the authors cut the $q$-exponents $0,\dots,2p-3$ into a lower range and an upper range of $p-1$ values each, so that $Y_{2p-3}(p,q)$ falls into two disjoint blocks. The lower block's subset sums are thick, that is, they contain arbitrarily long runs of consecutive integers; the authors get this from a complete residue system modulo $p$ found among the subset sums of $1,q,\dots,q^{p-2}$, together with a tiling lemma for one-dimensional self-affine sets. The upper block has finite deficit, so its subset sums have bounded gaps, and the sum of a thick set with a set of bounded gaps is cofinite; the two blocks being disjoint, no element is used twice.

Standing. The claimants are Zhao Song and Song Yue, who posted the result on the site's proof-claims page on 2026-09-04 under the forum name magiclinux; the claim's entry names the system ChatGPT 5.6 Sol. The site shows no verdict, the manuscript is not refereed, and no one has reviewed it, so the claim stays claimed. The problem itself is settled by Birch's theorem; this claim adds a quantitative bound.