Irreducibility of Littlewood polynomials of special degrees

Abstract

Let $f$ be sampled uniformly at random from the set of degree $n$ polynomials whose coefficients lie in $\{\pm 1\}$. A folklore conjecture, known to hold under GRH, states that the probability that $f$ is irreducible tends to $1$ as $n$ goes to infinity. We prove unconditionally that $$\underset{n\to \infty }{\mathrm{limsup}}\mathbb{P}(f\text{ is irreducible})=1.$$

4 pages; added case where n+1 goes through powers of 2; minor corrections