Perfect powers in values of certain polynomials at integer points

Shorey, T. N. (1986) Perfect powers in values of certain polynomials at integer points Mathematical Proceedings of the Cambridge Philosophical Society, 99 (2). pp. 195-207. ISSN 0305-0041

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Related URL: http://dx.doi.org/10.1017/S0305004100064112

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, ..., dt 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+dt)=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+di)≤k for 1 ≤ i ≤ t and hence t<α-1k+π(k).

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