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=1 k∏j=0 ℓxi,j n. We prove that the following are equivalent: (a) There are a real θ&gt;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=0 ℓai (2n) and s2n−1=∏i=0 ℓbi (2n−1) 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 c1x1 n+⋯+ckxk n is bounded, as n ranges over N+, if and only if x1=⋯=xk.

### 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=1 k∏j=0 ℓxi,j n. 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=0 ℓai (2n) and s2n−1=∏i=0 ℓbi (2n−1) 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 c1x1 n+⋯+ckxk n is bounded, as n ranges over N+, if and only if x1=⋯=xk.
##### Scheda breve Scheda completa Scheda completa (DC) 2018
2017
https://www.sciencedirect.com/science/article/pii/S0022314X17303943
Integer sequences; Number of distinct prime factors; S-unit equations; Sum of powers
Leonetti, P; Tringali, S
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11383/2142090`
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