Erdős Problem #119

Erdős problem #119 · erdosproblems.com/119

For unit-modulus complex numbers $z_i$ , let $p_n(z)=\prod_{i\le n}(z-z_i)$ and $M_n=\max_{|z|=1}|p_n(z)|$ . Erdős's prize question: is there $c>0$ with $\sum_{k\le n} M_k > n^{1+c}$ ?

Result: Proved
Field: Analysis, Polynomials
Posed by: Paul Erdős
Year posed: 1957
Years open: 69y
Solved: 2026-07
Model: GPT-5.6
Vendor: OpenAI
Collaborators: Samuel Korsky
Verification: Site-confirmed
Notability: No dedicated article

What the AI did

GPT-5.6, with Samuel Korsky, resolved Erdős's prize question, proving $\sum_{k\le n} M_k \gg n^{5/4}/\sqrt{\log n}$ (hence $M_n > n^{1/4-o(1)}$ infinitely often).

Verification

Marked solved on erdosproblems.com; carried an Erdős prize of USD 100. Resolved via a proof claim by GPT-5.6 and Samuel Korsky; not formally Lean-verified.

Source

erdosproblems.com