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Erdős Problem #539

Number theory · posed by 1973 · solved

2 comments

Statement

For ∣A∣=n|A| = n, how small can the cofactor set Q(A)={a/gcd⁡(a,b):a,b∈A}Q(A) = \{a / \gcd(a,b) : a, b \in A\} be? The answer is h(n)=n1/2+o(1)h(n) = n^{1/2 + o(1)}: a new upper bound h(n)≤n1/2exp⁡(O(log⁡n))h(n) \le n^{1/2} \exp(O(\sqrt{\log n})) matches the classical lower bound.

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

  1. construction · #1

    ProofCouncil (GPT-5.5 Pro)

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

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

    The upper-bound construction was found by the ProofCouncil harness running GPT-5.5 Pro.

    main exponent determined; sharper subpolynomial factors remain open

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

    lean: correctLean

    scope Lean formalization of the result

    Lean record alongside the official Erdős problems update marking the exponent determined.

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

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