Breakthrough Tracker record
Erdős Problem #421: a claimed density-one distinct-products construction
Chojecki gives a prime-gap greedy construction that claims to retain a density-one set while preventing equal products on distinct consecutive blocks. Sneiderman’s separate nine-page reconstruction fills in the cancellation, curve-counting and forest-summation steps and sharpens one rejected-gap estimate.
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- Stable ID
math-erdos-421-distinct-consecutive-products-2026- Revision
math-erdos-421-distinct-consecutive-products-2026.v1- Field
- Mathematics · Number theory
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Claimed proof with independent audit; specialist review pending
- Last checked
AI role
Chojecki’s preprint says GPT-5.6 Sol found the proof while iterating on the author’s earlier attempts. The audit discloses OpenAI Codex use for literature searches, exploratory work, exact computation, proof checking and drafting.
Record details
- Problem or result
- Erdős Problem #421, asking for a density-one increasing sequence in which all products over distinct consecutive index intervals are different
- Authors
- Przemek Chojecki; expanded audit and reconstruction by Rob Sneiderman
- Institutions
- Ulam AI; independent public GitHub repository
- Result date
- Proof preprint dated July 13; independent audit posted July 20, 2026
Why it matters
A correct construction would answer an Erdős–Graham question by improving the previously recorded density lower bound to natural density one.
Limits
Erdős Problems still labels the problem OPEN and explicitly says its proof-claim listing is not a correctness review. The proof and audit are self-published, not peer reviewed or on arXiv, and no specialist assessment was located. The audit repository is complete and its TeX rebuilt successfully in this maintenance cycle, but that establishes reproducibility of the document, not correctness of the mathematical argument.
Sources
- Primary: Przemek Chojecki, Distinct Consecutive Products
- Primary: Erdős Problems #421 status and proof-claim record
- Independent: Rob Sneiderman, pinned audit and reconstruction PDF
- Independent: Pinned source repository for the audit
Correction and revision history
- 2026-07-23 — Added provisionally after the complete five-page proof, a separate expanded audit, pinned TeX source and a successful local document rebuild became inspectable; no specialist acceptance is implied.
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