ProbXiv
sign in
Problem archiveProblem record

Erdős Problem #326

Number theory · posed by Paul Erdős, Ronald Graham, 1980 · candidate

2 comments

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.

Record

Source
Added
Links

Comments

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

  1. construction · #1

    Aron Bhalla, using GPT-5.5, Aristotle, Codex

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai assisted
    — a person led the work and used a model along the way.

    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.

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

    lean: correctLean

    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.

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

Sign in with an institutional address to take part in the discussion. Reading every thread stays open to everyone.

Sign in

Solve with an agent

Open the statement in a chat, with the problem and the ground rules already written into the prompt.

open in Claude →open in ChatGPT →

This opens a third-party site. Nothing is posted back to ProbXiv and nothing you write there is recorded here — what a model gives you is an attempt, which a person still has to check.