Ben Green's Open Problem 93

Is a random polynomial with coefficients in 0,1 and nonzero constant term almost surely irreducible? More precisely, writing p_n for the probability that 1 + a_1 x + ⋯ + a_n-1 x^n-1 + x^n is irreducible, where the a_i are independent and take the values 0 and 1 with equal probability, does p_n → 1?

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.

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Cite

@misc{cairn-green-93,
  title        = {Ben Green's Open Problem 93},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-93}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Is a random polynomial with coefficients in ${0, 1}$ and nonzero constant term almost surely irreducible?

More precisely, writing $p_{n}$ for the probability that $1 + a_{1} x + \dots + a_{n - 1} x^{n - 1} + x^{n}$ is irreducible, where the $a_{i}$ are independent and take the values $0$ and $1$ with equal probability, does $p_{n} \to 1$ ?

The probability $p_{n}$ is irreducibleProb 1 n. The conjecture is due to Odlyzko and Poonen [BrVa19, Section 1].

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«93».

theorem green_93 : Tendsto (fun n : ℕ ↦ irreducibleProb 1 n) atTop (𝓝 1)

What 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. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

[Gr24] Green, Ben. "100 open problems." (2024).
[BrVa19] Breuillard, E. and Varjú, P. P., *Irreducibility of random polynomials of large degree*. Acta Math. 223 (2019), 195--249. arXiv:1810.13360
[BSKK23] Bary-Soroker, L., Koukoulopoulos, D. and Kozma, G., *Irreducibility of random polynomials: general measures*. Invent. Math. 233 (2023), 1041--1120. arXiv:2007.14567
[Ko99] Konyagin, S. V., On the number of irreducible polynomials with 0,1 coefficients. Acta Arith. 88 (1999), 333--350.

Source and licence

Imported from Formal Conjectures (Ben Green's 100 open problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.