A new characterization for the Lucas-Carmichael Integers and sums of base-$p$ digits

TL;DR

This paper characterizes Lucas-Carmichael integers using base-$p$ digit sums, explores their properties, and shows infinitely many exist assuming the prime $k$-tuples conjecture.

Contribution

It provides a new necessary and sufficient condition for Lucas-Carmichael integers based on digit sums and links their infinitude to the prime $k$-tuples conjecture.

Findings

Characterization of Lucas-Carmichael integers via base-$p$ digit sums

Identification of interesting properties of these integers

Proof of infinitely many such integers assuming the prime $k$-tuples conjecture

Abstract

In this paper, we prove a necessary and sufficient condition for the Lucas-Carmichael integers in terms of the sum of base-$p$ digits. We also study some interesting properties of such integers. Finally, we prove that there are infinitely many Lucas-Carmichael integers assuming the prime $k$-tuples conjecture.