Three pearls of Bernoulli numbers

TL;DR

This paper explores three significant aspects of Bernoulli numbers, providing new insights and proofs related to longstanding questions and relationships in number theory and mathematics.

Contribution

It offers new responses to open problems posed by Carlitz and Apostol, and presents an original proof of a known relationship among Bernoulli numbers.

Findings

Answer to Carlitz's 1971 problem

Solution to Apostol's 2008 question

New proof of a Bernoulli number relationship

Abstract

The Bernoulli numbers are fascinating and ubiquitous numbers, they occur in several domains of Mathematics like Number theory (FLT), Group theory, Calculus and even in Physics. Since Bernoulli's work, they are yet studied to understand their deep nature and particularly to find relationships between them. In this paper, we give, firstly, a short response to a problem stated, in 1971, by Carlitz and studied by many authors like Prodinger \cite{PRO}, the second pearl is an answer to a question raised, in 2008, by Tom Apostol .The third pearl is another proof of a relationship already given in 2011, by the authors.