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Claim. Samuel Korsky's proof claim of 18 July 2026, which erdosproblems.com lists as a partial proof claim of Problem 424, claims a proof that the set of integers generated from $2$ and $3$ by appending $a_ia_j-1$ for all $i$ and $j$, so that $a_i^2-1$ is appended too, has positive lower density. The argument composes the affine maps $x\mapsto mx-1$ for the multipliers $2,3,5,8,9$, early members of that sequence, and uses a finite interval partition with transition probabilities to show that distinct compositions with a common slope land on distinct integers often enough. The claimant notes that this variant is the wording of Green's open problem list, while Erdős's own statement and erdosproblems.com require distinct factors. The proof is an external document linked from the tab.

Submission note. Posted to erdosproblems.com as a proof claim by Samuel Korsky (account SamKorsky) on 18 July 2026, giving "GPT 5.6-Pro" as the AI used:

I believe I am able to prove (with help from GPT-5.6 Pro) the positive lower density claim for a more favorable sequence (namely, removing the $i \ne j$ restriction). The general idea is to produce many distinct affine maps with a common slope (using initial elements of the sequence $2, 3, 5, 8, 9$) and use a finite interval partition with transition probabilities to distinguish the resulting maps. Comments are welcome! Notes: Some remarks with regards to the literature: [Er77c] states the problem exactly as written here, while the wording in Green's open problem list allows for a1 = a2 (the case studied here). However, given that Green cites this page and Erdos directly, I assume this difference was unintentional. Additionally, I spent some time prompting GPT 5.6-Pro about developing a similar argument for the exact problem here, but was only able to GPT to claim (not human-verified) a proof that if $A$ is the set corresponding to the sequence in the problem, then $|A \cap [1,x]| \gg \frac{x}{\sqrt{\log x}}$. If there's interest I can try to verify its correctness and write it up.

Why it is rejected. It answers a variant, not the Statement. Problem 424 appends $a_ia_j-1$ only for $i\neq j$, as Erdős's 1977 report and erdosproblems.com state it. The claim treats instead the sequence in which the two factors may coincide, whose set contains the problem's set and is strictly larger: it contains $8=3\cdot3-1$, which the problem's set lacks. Positive lower density of the larger set gives no lower bound for the smaller one, so its mathematics answers the variant and does not settle this problem; the value proved is its answer to that variant. The claimant's full claim of 20 July 2026, which extends the argument to distinct factors, is recorded on its own claim page.

Standing. Rejected, as answering a variant. erdosproblems.com lists the entry with two comments and no review, and this corpus has not read the external document.

Depends on. No page of this wiki.