Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients

Bernd C. Kellner

Abstract

It is well known that the Bernoulli polynomials $\mathbf{B}_n(x)$ have nonintegral coefficients for $n \geq 1$. However, ten cases are known so far in which the derivative $\mathbf{B}'_n(x)$ has only integral coefficients. One may assume that the number of those derivatives is finite. We can link this conjecture to a recent conjecture about the properties of a product of primes satisfying certain $p$-adic conditions. Using a related result of Bordellès, Luca, Moree, and Shparlinski, we then show that the number of those derivatives is indeed finite. Furthermore, we derive other characterizations of the primary conjecture. Subsequently, we extend the results to higher derivatives of the Bernoulli polynomials. This provides a product formula for these denominators, and we show similar finiteness results.

On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients

Abstract

It is well known that the Bernoulli polynomials have nonintegral coefficients for . However, ten cases are known so far in which the derivative has only integral coefficients. One may assume that the number of those derivatives is finite. We can link this conjecture to a recent conjecture about the properties of a product of primes satisfying certain -adic conditions. Using a related result of Bordellès, Luca, Moree, and Shparlinski, we then show that the number of those derivatives is indeed finite. Furthermore, we derive other characterizations of the primary conjecture. Subsequently, we extend the results to higher derivatives of the Bernoulli polynomials. This provides a product formula for these denominators, and we show similar finiteness results.
Paper Structure (5 sections, 13 theorems, 41 equations)

This paper contains 5 sections, 13 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Denominators and p-adic properties
  3. Denominators and higher derivatives
  4. Proofs of the theorems
  5. Acknowledgments

Key Result

Theorem 3

Conjecture conj:kel(i) and (ii) imply Conjecture conj:main(ii) and (i), respectively.

Theorems & Definitions (29)

  • Conjecture 1
  • Conjecture 2: Kellner Kellner:2017
  • Theorem 3
  • Corollary 4: Bordellès, Luca, Moree, and Shparlinski BLMS:2018
  • Theorem 5
  • Remark 6
  • Theorem 7: Kellner and Sondow KellnerSondow:2021
  • Corollary 8
  • proof
  • Theorem 9
  • ...and 19 more