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On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-$p$ digits

Bernd C. Kellner, Jonathan Sondow

Abstract

We give a new characterization of the set $\mathcal{C}$ of Carmichael numbers in the context of $p$-adic theory, independently of the classical results of Korselt and Carmichael. The characterization originates from a surprising link to the denominators of the Bernoulli polynomials via the sum-of-base-$p$-digits function. More precisely, we show that such a denominator obeys a triple-product identity, where one factor is connected with a $p$-adically defined subset $\mathcal{S}$ of the squarefree integers that contains $\mathcal{C}$. This leads to the definition of a new subset $\mathcal{C}'$ of $\mathcal{C}$, called the "primary Carmichael numbers". Subsequently, we establish that every Carmichael number equals an explicitly determined polygonal number. Finally, the set $\mathcal{S}$ is covered by modular subsets $\mathcal{S}_d$ ($d \geq 1$) that are related to the Knödel numbers, where $\mathcal{C} = \mathcal{S}_1$ is a special case.

On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-$p$ digits

Abstract

We give a new characterization of the set of Carmichael numbers in the context of -adic theory, independently of the classical results of Korselt and Carmichael. The characterization originates from a surprising link to the denominators of the Bernoulli polynomials via the sum-of-base--digits function. More precisely, we show that such a denominator obeys a triple-product identity, where one factor is connected with a -adically defined subset of the squarefree integers that contains . This leads to the definition of a new subset of , called the "primary Carmichael numbers". Subsequently, we establish that every Carmichael number equals an explicitly determined polygonal number. Finally, the set is covered by modular subsets () that are related to the Knödel numbers, where is a special case.

Paper Structure

This paper contains 8 sections, 14 theorems, 91 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Carmichael numbers and squarefree integers
  3. Bernoulli numbers and polynomials
  4. Polygonal numbers
  5. Proofs of Theorems 2.1, 2.2, and 2.3
  6. Proofs of Theorems 3.1 and 3.2 and Corollary 3.3
  7. Proofs of Theorems 4.1 and 4.2 and Corollary 4.3
  8. Modular properties of the set S

Key Result

Theorem \oldthetheorem

A composite number $m$ is a Carmichael number if and only if $m$ is squarefree and every prime divisor $p$ of $m$ satisfies $p - 1 \mid m - 1$.

Theorems & Definitions (28)

  • Theorem \oldthetheorem: Korselt's criterion Korselt:1899 (1899)
  • Theorem \oldthetheorem: Carmichael Carmichael:1910Carmichael:1912 (1910,1912)
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Remark \oldthetheorem
  • ...and 18 more