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Erdős Problem #326

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erdos-326Number theoryposed by Paul Erdős, Ronald Graham, 1980recorded: candidate

1 attempt · 1 machine check · no person has looked

Statement

Does there exist A={a1<a2<⋯ }⊂NA=\{a_1<a_2<\cdots\}\subset \mathbb{N} which is a minimal basis of order 22 (every large integer is the sum of 22 elements from AA, and no proper subset of AA has this property) such that lim⁡k→∞ak/k2=c\lim_{k\to \infty}a_k/k^2=c for some c≠0c\neq 0? A claimed construction gives a minimal basis with A(x)=Cx+O(1)A(x)=C\sqrt{x}+O(1), answering the question affirmatively; Erdős and Graham had conjectured a negative answer.

Context

Affirmative answer claimed, contrary to the negative answer Erdős and Graham conjectured; erdosproblems.com still lists the problem open

A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.

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Interest

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Attempts

1 attempt

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  • #1

    Attempt 1

    constructionGPT-5.5, Aristotle, Codex with Aron Bhalla ·
    AI involvement
    ai assisted
    — a person led the work and used a model along the way.
    models
    GPT-5.5, Aristotle, Codex
    people
    Aron Bhalla

    Per the author's disclosure, most of the mathematics is his own, with GPT-5.5 used to stress-test ideas, suggest revisions, identify gaps and write up some proofs; the solution was then formalized over several weeks with Aristotle, Codex and GPT-5.5 into a roughly 15,000-line Lean proof confirming all claims in the manuscript.

    Reviews

    0 human reviews · 1 machine check

    No person has reviewed this attempt. 1 machine check below — a machine check is not human verification.

    • Machine check · not human verification

      machine: correct

      Recorded from Lean ·

      scope Lean formalization of the result

      The author reports a ~15,000-line Lean formalization, type-checkable online, confirming all claims of the manuscript. It has not been independently audited for statement fidelity, and erdosproblems.com has not accepted the claim: the site's owner found the AI-written exposition hard to digest while stressing that this was not a correctness objection.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

    Endorsements

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    Discussion of this attempt

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Discussion

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