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

Abstract

Given k,ℓ∈N+, let xi,j be, for 1≤i≤k and 0≤j≤ℓ, some fixed integers, and define, for every n∈N+, sn:=∑i=1k∏j=0ℓxi,jnj. 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.(b)There do not exist non-zero integers a0,b0,…,aℓ,bℓ such that s2n=∏i=0ℓai(2n)i and s2n−1=∏i=0ℓbi(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 c1,x1,…,ck,xk∈N+, the number of distinct prime factors of c1x1n+⋯+ckxkn is bounded, as n ranges over N+, if and only if x1=⋯=xk.