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Claim. There is a set $A\subseteq\mathbb N$ of natural density one such that distinct consecutive blocks of its increasing enumeration $d_1<d_2<\cdots$ have distinct products (Theorem 1.1). This answers the question yes.

The result. P. Chojecki, Distinct consecutive products, preprint of 13 July 2026, posted in the site's thread that day and to arXiv as 2609.17543v1 on 14 July 2026 (math.NT). Library home: chojecki_2026_distinct_consecutive_products. The construction is greedy over prime gaps: start from $A_2=\{2\}$; for consecutive primes $p<q$, test the prefix $A_p\cup\{p+1,\ldots,q\}$, and if two distinct consecutive blocks of it have the same product keep only $q$, otherwise keep the whole interior. Every prime is retained, and each gap interior is wholly kept or wholly rejected. Lemma 2.1 shows every prefix, and $A$ itself, collision-free, and gives each rejected gap a canonical witness: an equality of products in which the later block is a full interval inside the rejected gap and the earlier block is longer. Rejected gaps form a forest by a chosen parent, the earlier rejected gap the witness's earlier block crosses. Parentless ("raw") short gaps are counted through uniform bounds for integral points on the split-product curves $\prod_i(x-a_i)=\prod_j(y+b_j)$, by the dimension-growth theorem of Castryck, Cluckers, Dittmann and Nguyen (Proposition 3.1, Lemma 3.2: $O(X^{4/5+o(1)})$ gaps); edges to a parent either share a multiplier or contract the scale from $X$ to $O(X^{20/39})$, and Proposition 4.3 bounds the total length of short rejected gaps by $O(X^{9/10+o(1)})$; gaps longer than $p^{1/20}$ have total length $o(X)$ by Li's theorem on primes in almost all short intervals (Lemma 2.3). Since the complement of $A$ is, apart from $1$, the union of rejected gap interiors, $A$ has density one.

History. Tao's thread comment of 18 October 2025 sketched the route: treat a collision as an integral point on a split-product curve, use Bombieri--Pila-type bounds for medium block lengths and random deletion for long ones, and identify the regime of block length of the order $\log N$ as the obstacle. Chojecki's write-up of 16 January 2026 pushed the determinant method into that regime without, by its own account, settling the problem; their claimed full proof of 17 January 2026, a dyadic deletion scheme with the algorithmic local lemma, and its modular rewrite of 19 January are recorded on their own page as a rejected claim: commenters found errors that the author acknowledged, and the proof of 13 July 2026 recorded here is a different, shorter argument. The preprint carries the author's disclosure on its first page: dated 13 July 2026, it says the proof was found by GPT-5.6 Sol while iterating on the author's previous attempts. The author's thread comment of the same day names GPT-5.6 Sol Ultra and says the disclaimer was added to the posted PDF at the curator's request; Pratt's exposition names the model as GPT-5.6 Sol. The curator noted that Pauwels's attempt of 10 July cited the same model family and arrived at a similar construction, and read the two as independent instances (see Pauwels's claim). Sneiderman's note of 21 July 2026 reproves the theorem on the same construction with a sharper exponent (Sneiderman's claim).

Acceptance. Reviewed: Kyle Pratt's A "digested" proof of Erdős Problem #421, a 19-page note (PDF dated 31 August 2026) posted as the site's proof exposition and last edited 1 September 2026, presents a complete proof of Theorem 1.1 whose presentation, the note says, follows the proofs of Chojecki and Sneiderman with minor modifications; its author takes responsibility for all mathematical claims and discloses that ChatGPT 5.6 Sol was used for understanding and analyzing the claimed proofs, and for exploration, wording and proofreading. The site labels the problem SOLVED, and the commentary of its curator, Thomas Bloom, who has no part in the proof (page last edited 1 September 2026), 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 a simplified exposition by Pratt. Not refereed: no journal publication was found on 2026-10-07, and the arXiv record has one version. Not formalized in this corpus: the Codex formalization of Sneiderman's write-up of this construction (Sneiderman's claim) was not built or audited here, and Sharma's Lean development is a different argument resting on three axioms (Sharma's claim). This corpus has not reviewed the proof: the library card digests the theorem and the lemma structure, and no step was checked. The proof's inputs from the literature are the dimension-growth bound of Castryck, Cluckers, Dittmann and Nguyen (Algebra Number Theory 14 (2020)) and Li's theorem on primes in almost all short intervals, which the preprint cites as an arXiv preprint of 2025; the author remarks in the thread that a weaker published result suffices for the latter.

Depends on. No page of this wiki.