Prime-valued polynomials

Abstract

In a recent Note Nick Lord proved that if a polynomial f(x) of degree n takes prime values for 2n + 1 integer values of x, then it is irreducible. (All polynomials discussed here are assumed to have integer coefficients and, like Lord, I regard numbers of the form –p where p is a positive prime as primes in their own right.) He wondered whether his result is the best possible one of its kind. If we restrict attention to positive primes, a best possible result can be achieved (Theorems 2 and 3 below). When we allow negative primes, Lord's result can be improved (see Theorems 1, 3 and 4) by an observation which still surprises me.