Perfect powers in values of certain polynomials at integer points

Abstract

1. For an integer v > 1, we define P ( v ) to be the greatest prime factor of v and we write P (1) = 1. Let m ≥ 0 and k ≥ 2 be integers. Let d 1 , …, d t with t ≥ 2 be distinct integers in the interval [1, k ]. For integers l ≥ 2, y > 0 and b > 0 with P ( b ) ≤ k , we consider the equation Put so that ½ < v t ≤ ¾. If α > 1 and k α < m ≤ k l , then equation (1) implies that for 1 ≤ i ≤ t and hence