Integer-valued Polynomial - Fixed Prime Divisors

Fixed Prime Divisors

Integer-valued polynomials may be used effectively to solve questions about fixed divisors of polynomials. For example, the polynomials P with integer coefficients that always take on even number values are just those such that P/2 is integer valued. Those in turn are the polynomials that may be expressed as a linear combination with even integer coefficients of the binomial coefficients.

In questions of prime number theory, such as Schinzel's hypothesis H and the Bateman–Horn conjecture, it is a matter of basic importance to understand the case when P has no fixed prime divisor (this has been called Bunyakovsky's property, for Viktor Bunyakovsky). By writing P in terms of the binomial coefficients, we see the highest fixed prime divisor is also the highest common factor of the coefficients in such a representation. So Bunyakovsky's property is equivalent to coprime coefficients.

As an example, the pair of polynomials n and n2 + 2 violates this condition at p = 3: for every n the product

n(n2 + 2)

is divisible by 3. Consequently there cannot be infinitely many prime pairs n and n2 + 2. The divisibility is attributable to the alternate representation

n(n + 1)(n − 1) + 3n.