Erdos's question on the maximum modulus of real Littlewood polynomials: is it at least (1 + c) sqrt(N)?

For signs $\epsilon_k\in\{-1,1\}$ let $P(z)=\sum_{k=0}^{N-1}\epsilon_k z^k$ . Parseval gives $\|P\|_{L^2(|z|=1)}=\sqrt N$ , so $\max_{|z|=1}|P(z)|\ge\sqrt N$ . Erdos asked whether this is far from sharp: is there an absolute constant $c>0$ with $\max_{|z|=1}|P(z)|\ge(1+c)\sqrt N$ for every such polynomial of large length? The analogue for complex unimodular coefficients was refuted by Kahane's ultraflat polynomials (1980); the Rudin-Shapiro polynomials give $O(\sqrt N)$ and Balister et al. gave two-sided flatness up to constants, but the constant $1+c$ for real signs stayed open. Is $\liminf_N \min_{\epsilon}\max_{|z|=1}|P(z)|/\sqrt N>1$ ?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Littlewood polynomials; extremal problems on the unit circle
Posed by: Paul Erdos
Year posed: 1957
Years open: 69y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Contested
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 40 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: $\min_{\epsilon}\max_{|z|=1}|\sum_{k<N}\epsilon_k z^k|=(1+o(1))\sqrt N$ as $N\to\infty$ through all integers, so no constant $c>0$ as in Erdos's question exists. The construction relaxes to real coefficients in $[-1,1]$ with small defect, using sampled quadratic-phase modes and a hypergraph interval packing, then rounds to signs by a Lovett-Meka partial-coloring argument. It is existential and gives no rate. The principal paper gives no uniform lower bound; the October 5 companions add $|P|\ge\sqrt N/16$ and then full two-sided ultraflatness $(1\pm\epsilon)\sqrt N$ at every large length.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript is dated September 23, 2026. Two companions dated October 5, 2026 strengthen it: one adds a lower bound of sqrt(N)/16, the other makes the polynomials two-sided ultraflat; the latter says it reuses lemmas of a 'version-2' refinement of the principal paper.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 ( $\lim_N m_N=1$ through all integers $N$ ) was read against Erdos's real-sign question; it answers it in the negative. lean/formalization.yaml lists a main result for this manuscript (comparator AsymptoticallyMinimalLittlewood, declaration OAI.AsymptoticallyMinimalLittlewood.main). The comparator statement was read here: for every $\eta>0$ there is $N_0$ such that for every $N\ge N_0$ there are real signs with $|\sum_{k<N}\epsilon_k z^k|\le(1+\eta)\sqrt N$ for all $|z|=1$ . This states the headline claim exactly. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The proof is existential with no rate or algorithm, as the paper says. The paper says its theorem conflicts with nonflatness claims in preprints of el Abdalaoui (arXiv:1609.03435 and two others) and examines specific issues in those arguments in its Appendix A; no public response to this release was found or searched for here. Listed as Contested because of the conflicting claim described in the claim issue.

Claim issue

This result conflicts with published claims. Preprints by el Abdalaoui (arXiv:1609.03435 and two others) claim the opposite, that such flat polynomials do not exist; the release's manuscript argues in its Appendix A that those arguments have specific gaps. Until the conflict is settled in public, the entry is Contested.

Erdos's question on the maximum modulus of real Littlewood polynomials: is it at least (1 + c) sqrt(N)?

What was actually shown

What the AI did

Verification

Claim issue

Sources

Discussion