Prime Numbers and Irreducible Polynomials

The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry. There are certain conjectures indicating that the connection goes well beyond analogy. For example, there is a famous conjecture of Buniakowski formulated in 1854 (see Lang [3, p. 323]), independently reformulated by Schinzel, to the effect that any irreducible polynomial f (x) in Z[x] such that the set of values f (Z+) has no common divisor larger than 1 represents prime numbers infinitely often. In this instance, the theme is to produce prime numbers from irreducible polynomials. This conjecture is still one of the major unsolved problems in number theory when the degree of f is greater than one. When f is linear, the conjecture is true, of course, and follows from Dirichlet’s theorem on primes in arithmetic progressions. It is not difficult to see that the converse of the Buniakowski conjecture is true; namely, if a polynomial represents prime numbers infinitely often, then it is an irreducible polynomial. To see this, let us try to factor f (x) = g(x)h(x) with g(x) and h(x) in Z[x] of positive degree. The fact that f (x) takes prime values infinitely often implies that either g(x) or h(x) takes the value ±1 infinitely often. This is a contradiction, for a polynomial of positive degree can take a fixed value only finitely often. There is a stronger converse to Buniakowski’s conjecture that is easily derived (see Theorem 1). To be specific, if a polynomial f (x) belonging to Z[x] represents a single prime number for some sufficiently large integer value of x , then the polynomial is irreducible. A classical result of A. Cohn (see Pölya and Szegö [5, p. 133]) states that, if we express a prime p in base 10 as