—that is, for it to be unfactorable into the product of lower-degree polynomials with integer coefficients.
The criterion is often stated as follows: The theorem can be generalized to other bases as follows: The base 10 version of the theorem is attributed to Cohn by Pólya and Szegő in Problems and Theorems in Analysis[2] while the generalization to any base b is due to Brillhart, Filaseta, and Odlyzko.
[3] It is clear from context that the "A. Cohn" mentioned by Polya and Szegő is Arthur Cohn (1894–1940), a student of Issai Schur who was awarded his doctorate from Frederick William University in 1921.
[4][5] A further generalization of the theorem allowing coefficients larger than digits was given by Filaseta and Gross.
[1] The converse of this criterion is that, if p is an irreducible polynomial with integer coefficients that have greatest common divisor 1, then there exists a base such that the coefficients of p form the representation of a prime number in that base.
This is the Bunyakovsky conjecture and its truth or falsity remains an open question.