Identities involving binomial-coefficients, Bernoulli- and Stirlingnumbers

Gottfried Helms

03´2007

A problem with powersums

Abstract:

a known problem of powersums is attacked. The problem is known as

"do any positive integer k and n exist so that 1^k+2^k+3^k+...+n^k = (n+1)^k ?"

The problem is here converted into an identity using Bernouli-numbers/ zeta-values and binomials

From a first impression this problem seemed to be easily solvable, but unfortunately it has more complexity than exhibited at the first glance... So a solution could not be given yet, but may be it gives a road, which can be continued meaningfully.

Contents:

1

Coefficients of polynomials f_k(x)

2

Real roots of the polynomials f_k(x)

3

Inverse of the matrix of coefficients of f_k(x)

Version:

05. Mrz 07

11:00