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

Abstract

Given $k,\ell \in {\mathbf{N}}^{+}$, let $x_{i,j}$ be, for $1\le i\le k$ and $0\le j\le \ell$, some fixed integers, and define, for every $n\in {\mathbf{N}}^{+}$, $s_n:={\sum }_{i=1}^k{\prod }_{j=0}^{\ell }{x}_{i,j}^{n^j}$. 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 $s_n$ is greater than the super-logarithm of $n$ to base $\theta$. (b) There do not exist non-zero integers $a_0,b_0,\dots ,a_{\ell },b_{\ell }$ such that $s_{2n}={\prod }_{i=0}^{\ell }{a}_i^{(2n{)}^i}$ and $s_{2n-1}={\prod }_{i=0}^{\ell }{b}_i^{(2n-1{)}^i}$ 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 $\mathcal{S}$, an effective bound on the number of non-degenerate solutions of an $\mathcal{S}$-unit equation in $k$ variables over the rationals) and the other using only elementary methods. As a corollary, we find that, for fixed $c_1,x_1,\dots ,c_k,x_k\in {\mathbf{N}}^{+}$, the number of distinct prime factors of $c_1{x}_1^n+\cdots +c_k{x}_k^n$ is bounded, as $n$ ranges over ${\mathbf{N}}^{+}$, if and only if $x_1=\cdots =x_k$.

arXiv comment10 pp., no figures. Fixed various mistakes around Lemma 2. To appear in Journal of Number Theory