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 $$\mathcal{D}=\{d_i{\}}_{i=0}^{\infty }\in \{0,\dots ,g-1{\}}^{\infty }$$ 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 $(d_0,\dots ,d_{N-1})$ 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.