Erdős Problem #522

Erdős problem #522 · erdosproblems.com/522

For $P_n(z) = \sum_{k=0}^n \varepsilon_k z^k$ with independent uniform signs, does the number $R_n$ of roots in $|z| \le 1$ satisfy $R_n/(n/2) \to 1$ almost surely? The manuscript proves the strong law with $R_n = n/2 + O_\omega(n^{149/150})$.

Result

Proved

Status

Candidate (review pending)

AI contribution

AI-discovered

Method

Argument

Field

Random Polynomials

Posed by

—

Year posed

1961

Years open

65y

Solved

2026-04-20

Model

GPT-5.5 Pro

Vendor

OpenAI

Collaborators

—

Verification

Unreviewed

Publication

Announced

Significance

10 / 100*

Disclosed cost

—

Wikipedia

No dedicated article