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 ⩾ 2 and then for every infinite sequence of g-ary digits we consider the counting function of integers n ⩽ N for which ∑^{n − 1}_{i = 0}d_igⁱ is prime. We construct sequences for which grows fast enough, and show that for some constant ϑ_g < g there are at most O(ϑ^N_g) initial elements (d₀, …, d_{N − 1}) of for which . We also discuss joint arithmetic properties of integers and mirror reflections of their g-ary expansions.

Keywords:

• digits
• primes

2000 AMS Subject Classification:

• 11A41
• 11A63
• 11N05

Acknowledgments

The authors are very grateful to Pieter Moree for introducing the question about the mirror primes, and also to Christian Mauduit and Joël Rivat for discussions of possible approaches to estimating M_g(N). The authors thankfully acknowledge the computer resources, technical expertise and assistance provided by the Santander Supercomputing services at the University of Cantabria.

Additional information

Funding

During the preparation of this paper, the first author was partially supported by project MTM2014-55421-P from the Ministerio de Economia y Competitividad and the second author was partially supported by Australian Research Council Grant DP140100118.