On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-$p$ digits
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2019
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911931058618368 |
|---|---|
| author | Kellner, Bernd C. Sondow, Jonathan |
| author_facet | Kellner, Bernd C. Sondow, Jonathan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1902_10672 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-$p$ digits Kellner, Bernd C. Sondow, Jonathan Number Theory 11B68, 11B83 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. |
| title | On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-$p$ digits |
| topic | Number Theory 11B68, 11B83 |
| url | https://arxiv.org/abs/1902.10672 |