ON THE FACTORIZATION OF f(n) FOR f(x) IN ℤ[x]

Abstract

Let S be a finite set of rational primes. For a non-zero integer n, define [Formula: see text], where |n|p is the usual p-adic norm of n. In 1984, Stewart applied Baker's theorem to prove non-trivial, computationally effective upper bounds for [n(n+1)⋯(n+k)]S for any integer k > 0. Effective upper bounds have also been given by Bennett, Filaseta, and Trifonov for [n(n + 1)]S and [n2 + 7]S, where S = {2, 3} and S = {2}, respectively. We extend Stewart's theorem to prove effective upper bounds for [f(n)]S for an arbitrary f(x) in ℤ[x] having at least two distinct roots.