# Definitions, notations and proofs for Bernoulli numbers

## TL;DR

This paper compiles definitions, notations, and elementary proofs for Bernoulli numbers, clarifying conventions and providing accessible proofs using basic combinatorial tools and symbolic notation.

## Contribution

It offers a unified presentation of Bernoulli number definitions and proofs, including elementary demonstrations for different conventions, using simple mathematical tools.

## Key findings

- Elementary proofs for Bernoulli numbers with different conventions
- Clarification of historical and modern notations
- Accessible derivations using binomial theorem and umbral notation

## Abstract

This is a collection of definitions, notations and proofs for the Bernoulli numbers $B_n$ appearing in formulas for the sum of integer powers, some of which can be found scattered in the large related historical literature in French, English and German. We provide elementary proofs for the original convention with ${\mathcal B}_1=1/2$ and also for the current convention with $B_1=-1/2$, using only the binomial theorem and the concise Blissard symbolic (umbral) notation.