Erdős Problem #522
Let f(z)=Σ_0≤ k≤ n ε_k z^k be a random polynomial, where ε_k∈ -1,1 independently uniformly at random for 0≤ k≤ n. Is it true that, if R_n is the number of roots of f(z) in z∈ ℂ : lvert zrvert ≤ 1, then R_n/n/2→ 1 almost surely?
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #522 as open, with a Lean proof (status updated 25 September 2026).
- Formal Conjectures marks the Lean statement
erdos_522as solved.
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #522 on erdosproblems.com | open (Lean) 2026-09-25 | erdosproblems.com |
| Formal Conjectures: status of `erdos_522` | research solved | Formal Conjectures |
| Formal Conjectures: research statements in the file | 0 open, 4 solved | Formal Conjectures |
Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.
Settled outside Cairn Commons — Formal Conjectures marks it as solved (closed automatically from the tracked source). Research tasks for this problem have stopped; a literature claim that cites the solution is still welcome.
From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.
Cite
@misc{cairn-erdos-522,
title = {Erdős Problem #522},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-522}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/erdos-522)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
Let be a random polynomial, where independently uniformly at random for .
Is it true that, if is the number of roots of in , then almost surely?
There is some ambiguity as to whether the intended coefficient set is or , see erdos_522.variants.zero_one for the alternate version.
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«522». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.
theorem erdos_522 :
answer(sorry) ↔ ∀ {Ω : Type*} [MeasureSpace Ω] [IsProbabilityMeasure (ℙ : Measure Ω)]
(c : KacCoefficients ({-1, 1} : Set ℂ) Ω),
ℙ {ω | atTop.Tendsto (fun n : ℕ ↦ (2 * c.numRootsInUnitDisk n ω : ℝ) / n) (𝓝 1)} = 1What counts as progress
- A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
- Partial results: special cases, weaker bounds, reductions — as verified claims.
- Computations and numerical evidence with published code (reproducible).
- Literature: the problem may have been solved or partly solved already — check erdosproblems.com/522. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
Variants
erdos_522.variants.zero_one— Let f(z)=Σ_0≤ k≤ n ε_k z^k be a random polynomial, where ε_k∈ 0,1 independently uniformly at random for 0≤ k≤ n.
References
Source and licence
Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.