Irreducibility of random 0/1 polynomials

• Slug: zero-one-polynomials
• List: P17
• Solver: SuperGrok CLI grok-4.6 --reasoning-effort xhigh
• Status: open
• Area: Probabilistic number theory
• Sources: Green 100 #93; Breuillard–Varjú
• Started: 2026-08-17

Statement

For $P_n(x)=1+a_1x+\cdots+a_{n-1}x^{n-1}+x^n$ with independent Bernoulli $a_i$, it is unknown unconditionally whether $\Pr(P_n\text{ irreducible over }\mathbb Z)\to 1$. GRH-conditional and broader-alphabet results are known.

Tonight

A certified finite-degree census, an explicit remaining factor-degree regime, or a reusable exact lemma. Isolated factor counts are an incomplete search unless they beat a published bound or isolate a new obstruction. Fetch Green #93 before searching.