Cohn's irreducibility criterion

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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.