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Nathanson's Problems on Product Intersection Sets

Number theory · posed by Melvyn B. Nathanson, 2026 · solved

2 comments

Statement

Nathanson asked which subsets of N\mathbb{N} can occur as product intersection sets of a family of semigroup subsets, for arbitrary and for decreasing families (his Problems 10 and 11). Both are solved by complete classifications.

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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. proof attempt · #1

    Harmonic Aristotle, with Wouter van Doorn, Pietro Monticone and Quanyu Tang

    The record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.

    AI involvement
    ai discovered
    — the result was found by a model.

    "Both classifications were autonomously discovered and formally verified in Lean by Aristotle." The appendix documents the prompts: Aristotle was asked to characterize the two cases separately, then combine them; it even adopted a stronger definition than the source paper's and the Lean code verifies the equivalence explicitly.

  2. Machine-checked by Lean on #1 · not a person

    lean: partially checkedLean

    scope Lean formalization of the core argument; statement correspondence not independently audited

    Discovered and kernel-checked in Lean by the same system; the human authors audited the informal-to-formal correspondence. Tier: Aristotle wrote both proof and formal statements (and at one point adopted a stronger definition than the source paper's, caught by the authors) - exactly the failure mode an independent statement audit exists for, and none has happened.

    Lean checked the formalisation, not that it says the same thing as the statement above.

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