On the number of distinct prime factors of a sum of super-powers

Given k, l is an element of N+, let x(i,j) be, for 1 <= i <= k and 0 <= j <= l, some fixed integers, and define, for every n is an element of N+, s(n) := Sigma(k)(i=1) Pi(l)(j=0) x(i,j)(nj). We prove that the following are equivalent: (a) There are a real theta > 1 and infinitely many indices n for which the number of distinct prime factors of sn is greater than the super-logarithm of n to base theta. (b) There do not exist non-zero integers a(0), b(0), ... , a(l), b(l) such that S-2n = Pi(l)(i=0) a(i)((2n)i) and s(2n - 1) = Pi(i=0l) b(i)((2n - 1)i) for all n. We will give two different proofs of this result, one based on a theorem of Evertse (yielding, for a fixed finite set of primes S, an effective bound on the number of non-degenerate solutions of an S-unit equation in k variables over the rationals) and the other using only elementary methods. As a corollary, we find that, for fixed c(1), x(1), ... , c(k), x(k) is an element of N+, the number of distinct prime factors of c(1)x(1)(n) + ... + c(k)x(k)(n) is bounded, as n ranges over N+, if and only if x(1) = ... = x(k) . (C) 2017 Elsevier Inc. All rights reserved.