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Erdős Problem #106

Geometry & topology · posed by Paul Erdős, 1932 · candidate

2 comments

Statement

If f(n)f(n) is the maximum total side length of nn interior-disjoint squares packed in the unit square, is f(k2+1)=kf(k^2 + 1) = k? An exact rational configuration packs 1717 squares with total side length greater than 44, refuting the identity at k=4k = 4.

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

  1. computation · #1

    Claude Opus 5

    No person is named on this work: the record names only the tool it came from, and no ProbXiv account is credited for it.

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

    The public Lean source credits Codex as formal author; The packing was discovered by Claude Opus 5 (Anthropic) against search infrastructure built and run by the submitter, and verified independently in exact rational arithmetic via separating-axis certificates.

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

    lean: correctLean

    scope Lean formalization of the result

    A 613-line, placeholder-free Lean proof of the counterexample; the erdosproblems.com page had not yet incorporated the result at audit time.

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

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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.