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 d₁, ..., 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 (m+d₁)...(m+d_t)=by¹. (1) Put v1=1/2(1+1/l−2), l=4,5,... so that ½ < vt ≤ ¾. If α > 1 and kα < m ≤ kl, then equation (1) implies that P(m+d_i)≤k for 1 ≤ i ≤ t and hence t<α^-1k+π(k).