Cohn's irreducibility criterion
From Number
Statement
Suppose 
is a polynomial with integer coefficients, i.e., 
. Suppose that all the coefficients of are nonnegative. Further, suppose 
is a natural number strictly greater than all coefficients. Then, if 
is a prime number, must be an irreducible polynomial.
An alternate formulation is as follows: for any , if a number with digits 
written in base 
is prime (so in particular 
for 
) then the polynomial 
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.
