# P2846: Irreducibility probability for random Littlewood polynomials

- ID: `P2846`
- Reference: `random-littlewood-irreducibility-limit`
- Page: https://theoremdb.org/statements/P2846
- Record maturity: Reviewed problem with recorded work

## 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 \(n\to\infty\).

### Problem setup

- **Definition.** A Littlewood polynomial has every coefficient in \(\{-1,1\}\); the displayed model fixes the leading coefficient to \(1\).
- **Definition.** Irreducible over \(\mathbb Q\) means that \(f_n\) cannot be written as a product of two positive-degree polynomials in \(\mathbb Q[X]\).
- **Remark.** The probability is taken over the \(2^n\) independent equally likely choices of the lower coefficients.

### What 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.

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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 \(n\ge N_0\).

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 \(n\ge N_0\).

### Background and intake notes

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.

- Original intake status: 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.
- 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: For \(n=1\), both possible polynomials \(X+1\) and \(X-1\) are irreducible.
- Recorded example: For \(n=2\), \(X^2-1\) occurs and is reducible, so the probability is not identically one at finite degree.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `random-littlewood-irreducibility-limit`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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. https://mathoverflow.net/questions/7969/irreducible-polynomials-with-constrained-coefficients
   - Also cited at See dataset.references[0] for the exact external source and locator.
   - Also cited at Editorial research route recorded 2026-07-31
   - forum; reference source; checked 2026-07-31
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>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. abstract and main theorems for Littlewood polynomials of special degree sequences https://arxiv.org/abs/2308.04878
   - preprint; reference source; arXiv:2308.04878, checked 2026-07-31; checked 2026-07-31
   - Source use: citation_only
   - 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.
3. <a id="reference-3"></a>Christian Borst, Evan Boyd, Claire Brekken, Samantha Solberg, Melanie Matchett Wood, and Philip Matchett Wood, “Irreducibility of Random Polynomials”. arXiv:1705.03709 (2017). Full preprint relevant to Irreducibility probability for random Littlewood polynomials. https://arxiv.org/abs/1705.03709
   - preprint; reference source; arXiv:1705.03709, checked 2026-07-31; checked 2026-07-31
   - Source use: citation_only
   - 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.