Faulhaber's Formula, Odd Bernoulli Numbers, and the Method of Partial Sums

TL;DR

This paper explores the relationship between Faulhaber's formula and odd Bernoulli numbers, providing a new proof of their connection using partial sums and induction, enhancing understanding of classical number theory results.

Contribution

The paper offers a novel proof of the converse relationship between Faulhaber's formula and odd Bernoulli numbers using partial sums and induction methods.

Findings

Confirmed that odd Bernoulli numbers are zero when Faulhaber's formula holds.

Provided a new proof of the converse relationship between Faulhaber's formula and Bernoulli numbers.

Enhanced understanding of the link between sum of powers and Bernoulli numbers.

Abstract

Let ``Faulhaber's formula'' refer to an expression for the sum of powers of integers written with terms in n(n+1)/2. Initially, the author used Faulhaber's formula to explain why odd Bernoulli numbers are equal to zero. Next, Cereceda gave alternate proofs of that result and then proved the converse, if odd Bernoulli numbers are equal to zero then we can derive Faulhaber's formula. Here, the original author will give a new proof of the converse using the method of partial sums and mathematical induction.