Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 421

claims/: The 6 claim pages of Problem 421, one per claimant's result; the problem's standing derives from them.

Statement. Is there a sequence $1\leq d_1<d_2<\cdots$ with density $1$ such that all products $\prod_{u\leq i\leq v}d_i$ are distinct?

Status. Solved: the answer is yes. The site labels the problem SOLVED (page last edited 1 September 2026). The result is Theorem 1.1 of Chojecki's preprint of 13 July 2026 (arXiv 2609.17543, 14 July 2026), a gap-greedy construction over consecutive primes whose proof the preprint states was found by GPT-5.6 Sol; Sneiderman's note of 21 July 2026 reproves it with a sharper exponent, and Pratt's digested proof of 1 September 2026, posted as the site's proof exposition, presents the argument in full following both. The site's curator records the answer as yes and describes its provenance as an approach sketched by Tao and fleshed out by Chojecki, full proofs claimed from GPT by several people independently, and Pratt's simplified exposition. No refereed publication exists as of 2026-10-07, and no formal proof has been built in this corpus; three further AI-assisted claims stand unreviewed, and Chojecki's claimed proof of January 2026 is recorded as rejected. Claim pages: Chojecki and Sneiderman (accepted); Kielhorn, Pauwels and Sharma (claimed); Chojecki's January claim (rejected). Selfridge's construction, recorded under Problem 786, gives such a sequence of density greater than $1/e-\epsilon$ for every $\epsilon>0$ .

Source. erdosproblems.com/421, accessed 2026-09-04, 2026-09-05 and 2026-10-07; the problem page carries a 47-comment thread and two proof claims. Cite as: T. F. Bloom, Erdős Problem #421, https://www.erdosproblems.com/421.

References.

[ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), p. 84. Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
[Ch26] Chojecki, P., Distinct consecutive products. Preprint of 13 July 2026; arXiv:2609.17543 (14 July 2026). Library home: chojecki_2026_distinct_consecutive_products.
[Ch26a] Chojecki, P., claimed proof of 17 January 2026 and its rewrite of 19 January 2026, PDFs served by the author's organization; rejected on acknowledged errors and superseded by [Ch26]. Not held.
[Pr26] Pratt, K., A "digested" proof of Erdős Problem #421. Note of 19 pages posted as the site's proof exposition (last edited 1 September 2026). Not held.
[Sn26] Sneiderman, R., Erdős Problem 421: audit and reconstruction. Note in a GitHub repository, 21 July 2026. Not held.
[Sh26] Sharma, A., Erdős Problem 421. GitHub repository with a Lean development, first posted 28 June 2026; proof claim of 26 July 2026. Not held.
[Ki26] Kielhorn, R., Distinct consecutive products in a density-one set via prime-gap deletions. Zenodo preprint, 10 July 2026, doi:10.5281/zenodo.21287065. Not held.
[Pa26] Pauwels, T., proof attempt, Overleaf document (main version of 10 July 2026 by the author's statement). Not held.

Formalization. Statement in formal-conjectures. Sharma's Lean development (see the claim page) proves its statement from three axioms standing for literature results absent from Mathlib; formal-conjectures links, as its formal proof, a Codex development in Boris Alexeev's repository that formalizes Sneiderman's write-up and reports only the standard axioms. Neither development was built in this corpus.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (9)

problems/integer_sequences/E0421/claims Sequences and Densities of Integers Sequences and Densities of Integers integer_sequences/chojecki_2026_distinct_consecutive_products Lemma 2.1 (p. 2): every prefix of the gap-greedy set is collision-free, and a rejected gap has a witness whose later side is an interval inside it Lemma 2.2 (p. 2): parent-child alternatives, equal or unequal, in the forest of rejected gaps Lemma 2.3 (p. 2): prime gaps longer than p^{1/20} have negligible total length Proposition 4.3 (p. 4): the short rejected gaps up to X have total length O(X^{9/10+o(1)})Theorem 1.1 (p. 1): a set of natural density one whose distinct consecutive blocks have distinct products

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