Erdős #1117 on maximum-modulus points answered negatively with GPT-6 help
Awaiting review
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 17 October 2026
Your AI and this story
- GPT-6 Astra163 days after its cutoff
- Claude Opus 5.5102 days after its cutoff
- Gemini 4 Argon193 days after its cutoff
- Grok 4.7132 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 102 days before it.
Key facts
- Paper: arXiv 2610.15067 [math.CV], ‘Maximum-modulus points on concentric circles’, single author Yunpeng Li, School of Data Science, CUHK Shenzhen; v1 Sat 10 Oct 2026 (16 KB), v2 Wed 14 Oct 2026 (14 KB; v2 replaces a ‘global generic-degree argument of version 1’ with a local one)
- Result: write f(z) = z^m g(z), g(0) ≠ 0, and let k be the first positive exponent with a nonzero Taylor coefficient of g; then {r > 0 : ν_f(r) > 2k} is locally finite in (0, ∞), the constant 2k is sharp, and lim inf ν_f(r) ≤ 2k < ∞ with no growth assumption
- Problem history (paper and erdosproblems.com/1117): Erdős asked around 1964 whether lim sup or lim inf of ν_f(r) can be infinite; Hayman’s problem list (Problem 2.16) records it; Herzog and Piranian answered the lim sup question yes in 1968; Glücksam and Pardo-Simón gave an ‘approximate’ affirmative answer to the lim inf question; the second question is also attributed to Clunie
- AI disclosure (paper, verbatim): ‘Assistance disclosure. OpenAI’s ChatGPT and Codex (GPT-6) were used to develop the proof and prepare the manuscript. The human author takes full responsibility for the mathematical arguments and the final manuscript.’
- Status on Oct 17, 2026: erdosproblems.com/1117 (last edited 29 Dec 2025) still lists the second question as open and shows one ‘proof claim’; no formalization; unrefereed
- Same author earlier posted a 72-page preprint on Quillen’s local parameter conjecture in dimension four (arXiv 2610.13469), listed in the 2026-10-09 catalogue
What happened
Yunpeng Li of the School of Data Science at CUHK Shenzhen posted a short paper (v1 on Oct 10, 2026, revised Oct 14) that answers the second question of Erdős Problem #1117. For an entire function f that is not a monomial, let ν_f(r) count the points on the circle |z| = r where |f| reaches its maximum. Herzog and Piranian showed in 1968 that ν_f(r) can be unbounded along some radii. Erdős’s second question, whether a single function can have ν_f(r) → ∞ on all large circles, stayed open. Li shows the answer is no: the count is bounded by 2k, where k is the gap between the first two nonzero Taylor terms of f, except on a locally finite set of radii, and the bound is sharp.
The paper’s closing disclosure says ChatGPT and Codex running GPT-6 were used “to develop the proof and prepare the manuscript”. It does not say which steps the models found.
Why it matters
It is another decades-old Erdős question (the list carries Hayman’s 1967/1974 problem-book provenance) answered in a preprint whose author credits a GPT-6 model with developing the proof. As with most such claims this month, it is unrefereed and erdosproblems.com has not yet updated the problem’s status.
What is disputed or not yet verified
| Verification | Unrefereed preprint (two versions); no formal proof; not yet marked solved on erdosproblems.com |
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Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 1 primary, 1 reaction
Primary
Reactions
- Erdős Problems #1117 (T. F. Bloom)erdosproblems.com, discussion
Changes
- Filed (read the paper’s text, its arXiv abstract page and erdosproblems.com/1117)