Arithmetic Properties of Integers in Chains and Reflections of $g$-ary Expansions

Abstract

Recently, there has been a sharp rise of interest in properties of digits primes. Here we study yet another question of this kind. Namely, we fix an integer base $g \ge 2$ and then for every infinite sequence $${\mathcal D} = \{d_i\}_{i=0}^\infty \in \{0, \ldots, g-1\}^\infty $$ of $g$-ary digits we consider the counting function $\varpi_{{\mathcal D},g}(N)$ of integers $n \le N$ for which $\sum_{i=0}^{n-1} d_i g^i$ is prime. We construct sequences ${\mathcal D}$ for which $\varpi_{{\mathcal D},g}(N)$ grows fast enough, and show that for some constant $\vartheta_g< g$ there are at most $O(\vartheta_g^N)$ initial elements $(d_0, \ldots, d_{N-1})$ of ${\mathcal D}$ for which $\varpi_{{\mathcal D},g}(N)=N+O(1)$. We also discuss joint arithmetic properties of integers and mirror reflections of their $g$-ary expansions.