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

TL;DR

This paper introduces a novel $p$-adic characterization of Carmichael numbers linked to Bernoulli polynomial denominators, revealing their connection to polygonal numbers and modular subsets related to Kn"odel numbers.

Contribution

It provides a new $p$-adic perspective on Carmichael numbers, introduces primary Carmichael numbers, and establishes their relation to polygonal numbers and modular subsets.

Findings

Carmichael numbers are characterized via Bernoulli polynomial denominators.

Every Carmichael number can be expressed as a polygonal number.

The set of special subsets $ extit{S}$ is covered by modular subsets related to Kn"odel numbers.

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 Kn\"odel numbers, where $\mathcal{C} = \mathcal{S}_1$ is a special case.