Irreducibility of Random {0,1} Polynomials

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For each integer $n\geq2$, let $\xi_1,\ldots,\xi_{n-1}$ be independent random variables with $\Pr(\xi_j=0)=\Pr(\xi_j=1)=1/2$, and set \[ F_n(T)=T^n+\sum_{j=1}^{n-1}\xi_jT^j+1\in\mathbb Z[T]. \] This is the uniform distribution on degree-$n$ polynomials with coefficients in $\{0,1\}$ and nonzero constant coefficient; the leading and constant coefficients are necessarily $1$. Irreducible means having no factorization into two polynomials of positive degree in $\mathbb Q[T]$.

Statement recorded in the supplied source.

Does \[ \lim_{n\to\infty}\Pr\left(F_n\text{ is irreducible over }\mathbb Q\right)=1? \] Here "almost surely" means probability tending to one as the degree tends to infinity, rather than an assertion about every term of one infinite random sequence. [S2]

Short English statement

Is a uniformly random binary-coefficient polynomial with constant coefficient one irreducible with probability tending to one as its degree grows?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 146 (GREEN-058). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Lior Bary-Soroker, Dimitris Koukoulopoulos and Gady Kozma, Irreducibility of random polynomials: general measures, arXiv:2007.14567v3 (2023), §1, discussion of binary coefficients and Theorem 1 with coefficient set $\{0,1\}$. https://arxiv.org/abs/2007.14567.
• S3 Ben Green, 100 Open Problems, maintained author list, Problem 93 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.