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Erdős Problem #848

Number theory · posed by Paul Erdős, András Sárközy, 1992 · solved

2 comments

Statement

Is the maximum size of a set A⊆{1,…,N}A\subseteq \{1,\ldots,N\} such that ab+1ab+1 is never squarefree (for all a,b∈Aa,b\in A) achieved by taking those n≡7(mod25)n\equiv 7\pmod{25}? Resolved for all sufficiently large NN: any near-maximal AA is contained in {n≡7(mod25)}\{n\equiv 7\pmod{25}\} or {n≡18(mod25)}\{n\equiv 18\pmod{25}\}, leaving only a finite check.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Mehtaab Sawhney and Mark Sellke, using GPT-5

    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 co developed
    — a person and a model developed the result together.

    Sawhney's note resolving the problem cites a ChatGPT (GPT-5) session in the provenance of the key lemma, and Tao's AI-contributions ledger records the solve as GPT-5 working with Sawhney and Sellke (October-November 2025).

    Resolved for all sufficiently large N via a stability theorem; small N remain a finite computation (erdosproblems.com marks the problem DECIDABLE)

  2. Recorded elsewhere on #1 · not checked here

    recorded: correctVibeMathed site check

    scope Reproduction by the VibeMathed site

    erdosproblems.com marks the problem resolved up to a finite check and links Sawhney's note; no formal artifact and no journal review.

    Repeated from the source; nothing was checked here.

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