[#P2846] Irreducibility probability for random Littlewood polynomials

A flat mathematical diagram showing random sign coefficients and roots of a Littlewood polynomial. — A schematic view of random sign coefficients and roots of a Littlewood polynomial.

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Problem. For each \(n\ge1\), choose independent random signs \(\varepsilon_0,\ldots,\varepsilon_{n-1}\in\{-1,1\}\) uniformly and set \(f_n(X)=X^n+\sum_{j=0}^{n-1}\varepsilon_jX^j\). Prove that \(\Pr(f_n\text{ is irreducible over }\mathbb Q)\to1\) as \(n\to\infty\).

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Definitions and notation

1Context

Each degree supplies a finite exact ensemble, so factor counts and local obstruction statistics can be pooled across independent runs. The theorem asks for uniform control of the vanishing reducible fraction.

2Problem setup

Definition 1 (A Littlewood polynomial has every coefficient in \(\{-1,1\}\); the displayed model fixes the leading coefficient to \(1\). A Littlewood polynomial has every coefficient in \(\{-1,1\}\); the displayed model fixes the leading coefficient to \(1\).

Definition 2 (Irreducible over \(\mathbb Q\). Irreducible over \(\mathbb Q\) means that \(f_n\) cannot be written as a product of two positive-degree polynomials in \(\mathbb Q[X]\).

Definition 3 (The probability). The probability is taken over the \(2^n\) independent equally likely choices of the lower coefficients.

Remark 1. Each degree supplies a finite exact ensemble, so factor counts and local obstruction statistics can be pooled across independent runs. The theorem asks for uniform control of the vanishing reducible fraction.

3What counts as a solution

Prove that for every \(\eta>0\) there is \(N_0\) such that \(\Pr(f_n\text{ is irreducible over }\mathbb Q)>1-\eta\) for every \(n\ge N_0\).
The proof must cover all degrees and all possible rational factor types. Conditional results must state their hypotheses and do not meet the unconditional target.

1Status

What counts as a solution

Current status (Current status and unresolved remainder). UNKNOWN as of 2026-07-31. The MathOverflow thread has no accepted proof of the limit. Bary-Soroker, Koukoulopoulos, and Kozma prove the limit along special degree sequences under stated hypotheses and obtain an unconditional limsup result, while the full all-degree limit remains described as conjectural in the checked literature. Prove that for every \(\eta>0\) there is \(N_0\) such that \(\Pr(f_n\text{ is irreducible over }\mathbb Q)>1-\eta\) for every \(n\ge N_0\).^[1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
UNKNOWN as of 2026-07-31. The MathOverflow thread has no accepted proof of the limit. Bary-Soroker, Koukoulopoulos, and Kozma prove the limit along special degree sequences under stated hypotheses and obtain an unconditional limsup result, while the full all-degree limit remains described as conjectural in the checked literature. Prove that for every \(\eta>0\) there is \(N_0\) such that \(\Pr(f_n\text{ is irreducible over }\mathbb Q)>1-\eta\) for every \(n\ge N_0\).	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 UNKNOWN as of 2026-07-31. The MathOverflow thread has no accepted proof of the limit. Bary-Soroker, Koukoulopoulos, and Kozma prove the limit along special degree sequences under stated hypotheses and obtain an unconditional limsup result, while the full all-degree limit remains described as conjectural in the checked literature. Prove that for every \(\eta>0\) there is \(N_0\) such that \(\Pr(f_n\text{ is irreducible over }\mathbb Q)>1-\eta\) for every \(n\ge N_0\).^[1] Relevance to this problem For random littlewood irreducibility limit, pins the dated research frontier: UNKNOWN as of 2026-07-31. The MathOverflow thread has no accepted proof of the limit. Bary-Soroker, Koukoulopoulos, and Kozma prove the limit along special degree sequences under stated hypotheses and obtain an unconditional limsup result, while the full all-degree limit remains described as. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details UNKNOWN as of 2026-07-31. The MathOverflow thread has no accepted proof of the limit. Bary-Soroker, Koukoulopoulos, and Kozma prove the limit along special degree sequences under stated hypotheses and obtain an unconditional limsup result, while the full all-degree limit remains described as conjectural in the checked literature. A complete resolution must satisfy this condition: Prove that for every \(\eta>0\) there is \(N_0\) such that \(\Pr(f_n\text{ is irreducible over }\mathbb Q)>1-\eta\) for every \(n\ge N_0\). Open full record →
Prove that for every \(\eta>0\) there is \(N_0\) such that \(\Pr(f_n\text{ is irreducible over }\mathbb Q)>1-\eta\) for every \(n\ge N_0\).	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove that for every \(\eta>0\) there is \(N_0\) such that \(\Pr(f_n\text{ is irreducible over }\mathbb Q)>1-\eta\) for every \(n\ge N_0\).^[1] Relevance to this problem For Irreducibility probability for random Littlewood polynomials, record random-littlewood-irreducibility-limit-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove that for every \(\eta>0\) there is \(N_0\) such that \(\Pr(f_n\text{ is irreducible over }\mathbb Q)>1-\eta\) for every \(n\ge N_0\). Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove that for every \(\eta>0\) there is \(N_0\) such that \(\Pr(f_n\text{ is irreducible over }\mathbb Q)>1-\eta\) for every \(n\ge N_0\). Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. UNKNOWN as of 2026-07-31. The MathOverflow thread has no accepted proof of the limit. Bary-Soroker, Koukoulopoulos, and Kozma prove the limit along special degree sequences under stated hypotheses and obtain an unconditional limsup result, while the full all-degree limit remains described as conjectural in the checked literature.

On 2026-07-27 all four MathOverflow answers and their comments were checked. They discuss cyclotomic factors, finite-field reductions, and earlier positive lower bounds without proving convergence to one for every degree.
Bary-Soroker, Koukoulopoulos, and Kozma, arXiv:2308.04878 and the 2025 IMRN publication, prove irreducibility results for special degrees and report the all-degree limit as the folklore conjecture.
The monic convention removes a harmless global sign. Constant term \(\pm1\) already rules out a zero root, but cyclotomic and noncyclotomic factors both require control.
Exact factorization counts by degree, factor type, and residue reductions are reusable. A Monte Carlo estimate alone cannot prove the limit.
Trap: proving \(\limsup=1\), a positive lower bound, or convergence along a subsequence leaves the displayed limit unresolved.

Recorded example 1. For \(n=1\), both possible polynomials \(X+1\) and \(X-1\) are irreducible.

Recorded example 2. For \(n=2\), \(X^2-1\) occurs and is reducible, so the probability is not identically one at finite degree.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemIrreducibility probability for random Littlewood polynomials

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Irreducibility probability for random Littlewood polynomials.” TheoremDB. P2846. Problem statement; statement text SHA-256 c817331631f3e755ed0d71c2533023b20daa5ec2b6aed51aaf6d3637dad8a24d. https://theoremdb.org/statement/?ref=P2846

BibTeX

@misc{theoremdb-problem-c817331631f3e755ed0d71c2533023b20daa5ec2b6aed51aaf6d3637dad8a24d,
  title = {{Irreducibility probability for random Littlewood polynomials}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 c817331631f3e755ed0d71c2533023b20daa5ec2b6aed51aaf6d3637dad8a24d},
  url = {https://theoremdb.org/statement/?ref=P2846}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net^[1], current as of July 31, 2026.

1References

Packet source. Irreducible polynomials with constrained coefficients, MathOverflow question 7969. Original CC0 probability formulation written after reading all four answers and comments and checking the recent special-degree theorem. mathoverflow.net checked 2026-08-01. Original CC0 probability formulation written after reading all four answers and comments and checking the recent special-degree theorem. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow thread has no accepted proof of the limit. Bary-Soroker, Koukoulopoulos, and Kozma prove the limit along special degree sequences under stated hypotheses and obtain an unconditional limsup result, while the full all-degree limit remains described as conjectural in the checked literature.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Irreducibility probability for random Littlewood polynomials: UNKNOWN as of 2026-07-27. The MathOverflow thread has no accepted proof of the limit. Bary-Soroker, Koukoulopoulos, and Kozma prove the limit along special degree sequences under stated hypotheses and obtain an unconditional limsup result, while the full all-degree limit remains described as conjectural in the checked literature.Source named by the research packet.
Lior Bary-Soroker, David Hokken, Gady Kozma, and Bjorn Poonen, “Irreducibility of Littlewood Polynomials of Special Degrees,” International Mathematics Research Notices 2025(21) (2025), article rnaf326. DOI 10.1093/imrn/rnaf326. Status evidence identified in the source record and checked at the linked publication. ↗preprint · primary source · arXiv:2308.04878, checked 2026-07-31 · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow thread has no accepted proof of the limit. Bary-Soroker, Koukoulopoulos, and Kozma prove the limit along special degree sequences under stated hypotheses and obtain an unconditional limsup result, while the full all-degree limit remains described as conjectural in the checked literature.Also cited at abstract and main theorems for Littlewood polynomials of special degree sequences.Source used to assess the problem's recorded status.For Irreducibility probability for random Littlewood polynomials, this source proves irreducibility limits along special degree sequences and leaves the all-degree limit unresolved.
Christian Borst, Evan Boyd, Claire Brekken, Samantha Solberg, Melanie Matchett Wood, and Philip Matchett Wood, “Irreducibility of Random Polynomials”. arXiv:1705.03709 (2017). Status evidence identified in the source record and checked at the linked publication. ↗preprint · primary source · arXiv:1705.03709, checked 2026-07-31 · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow thread has no accepted proof of the limit. Bary-Soroker, Koukoulopoulos, and Kozma prove the limit along special degree sequences under stated hypotheses and obtain an unconditional limsup result, while the full all-degree limit remains described as conjectural in the checked literature.Also cited at Full preprint relevant to Irreducibility probability for random Littlewood polynomials.Source used to assess the problem's recorded status.For Irreducibility probability for random Littlewood polynomials: UNKNOWN as of 2026-07-27. The MathOverflow thread has no accepted proof of the limit. Bary-Soroker, Koukoulopoulos, and Kozma prove the limit along special degree sequences under stated hypotheses and obtain an unconditional limsup result, while the full all-degree limit remains described as conjectural in the checked literature.

Original CC0 textbook restatement.

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[#P2846] Irreducibility probability for random Littlewood polynomials

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2Problem setup

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Recent contributions

2See also

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Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

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