Cohn's irreducibility criterion

Statement

Suppose $p$ is a polynomial with integer coefficients, i.e., $p (x) \in Z [x]$ . 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.