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

Abstract

Abstract. Given k, ℓ ∈ N + , let x i,j be, for 1 ≤ i ≤ k and 0 ≤ j ≤ ℓ, some fixed integers, and define, for every n ∈ N + , sn := k i=1 ℓ j=0 x n j i,j . We prove that the following are equivalent: (a) There are a real θ > 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 θ. 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 ∈ N + , the number of distinct prime factors of c 1 x n 1 + · · · + c k x n k is bounded, as n ranges over N + , if and only if x 1 = · · · = x k .