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OpenAI reports: Erdős’s reciprocal-sum progression conjecture

OpenAI reports that every positive-integer set with divergent reciprocal sum contains arithmetic progressions of every finite length, through fixed-length bounds r_k(N)≤C_k N exp(−c_k(log N)^ε_k).

Sources for this record

What was already known

Positive density guarantees from Szemerédi’s theorem do not automatically cover every set with divergent reciprocal sum. The needed quantitative bound addresses that stronger target.

What changed and why it matters

Extends progression existence to the divergent-reciprocal-sum target through the stated quantitative bound.

The AI contribution

OpenAI attributes these manuscripts to its internal model. The entry does not infer individual human contributions beyond that publisher attribution.

Limitations

Constants and the exponent depend on the fixed progression length and are not optimized. The claim does not provide a uniform bound as the length grows. Preprint. This is a source-statement and significance assessment, not a proof audit. No independent checking or successful formal compilation is asserted by this entry.

Verification evidence

Evidence labels show what supports the claim. Importance rates its impact if correct.

Author claim

The publisher reports the scoped result. AIProofWatch has not independently validated its correctness.

Accessible artifact / proof

The original written argument is accessible. Availability does not establish a successful proof check or Lean verification.

Correction and publication history

2026-10-11 · publication

Historical report added after statement, scope, and importance review. No proof audit or formal checking asserted; subscriber alerts disabled.

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