Cohn's irreducibility criterion

Statement

Suppose $p$ is a polynomial with integer coefficients, i.e., $p(x)\in \mathbb {Z} [x]$. Suppose that all the coefficients of $p$ are nonnegative. Further, suppose $b\geq 2$ is a natural number strictly greater than all coefficients. Then, if $p(b)$ is a prime number, $p$ must be an irreducible polynomial.

An alternate formulation is as follows: for any $b\geq 2$, if a number with digits $a_{n}a_{n-1}\dots a_{1}a_{0}$ written in base $b$ is prime (so in particular $0\leq a_{i}\leq b-1$ for $i\in \{0,1,\dots ,n\}$) then the polynomial $a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{1}x+a_{0}$ is irreducible.

Related facts

• Bunyakovsky conjecture is a conjectured converse of sorts: if a polynomial is irreducible and the set of its values does not have a gcd, then the polynomial must take prime values at infinitely many natural numbers.