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

Sridhar Tamilvanan, Subramani Muthukrishnan

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.

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

Abstract

In this paper, we prove a necessary and sufficient condition for the Lucas-Carmichael integers in terms of the sum of base- digits. We also study some interesting properties of such integers. Finally, we prove that there are infinitely many Lucas-Carmichael integers assuming the prime -tuples conjecture.
Paper Structure (4 sections, 20 theorems, 40 equations)

This paper contains 4 sections, 20 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lucas-Carmichael integers
  4. Some general forms of Lucas-Carmichael integers

Key Result

Theorem 1.1

(Korselt's criterion). A composite number $m$ is a Carmichael number if and only if $m$ is square-free and every prime divisor $p$ of $m$ satisfies $p - 1$$|$$m - 1.$

Theorems & Definitions (40)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.3
  • Lemma 2.1
  • proof
  • Corollary 2.1
  • proof
  • ...and 30 more