arXiv subjects

Maria S. Nogin

Publications and source records attributed to Maria S. Nogin.

2 recordsLinked to original sources

arXiv2017

Congruences of Power Sums

The following congruence for power sums, $S_n(p)$, is well known and has many applications: $1^n+2^n +\dots +p^n \equiv\begin{cases} -1 \text{ mod } p, & \text{ if } \ p-1 \ | \ n; 0 \text{ mod } p, & \text{ if } \ p-1 \ \not| \ n, \end{cases}$ where $n\in{\mathbb N}$ and $p$ is prime. We extend this congruence, in particular, to the case when $p$ is any power of a prime. We also show that the sequence $(S_n(m) \text{ mod } k )_{m \geq 1}$ is periodic and determine its period.

math.NT↗

arXiv2017

A Proof of Symmetry of the Power Sum Polynomials using a Novel Bernoulli Numbers Identity

The problem of finding formulas for sums of powers of natural numbers has been of interest to mathematicians for many centuries. Among these is Faulhaber's well-known formula expressing the power sums as polynomials whose coefficients involve Bernoulli numbers. In this paper we give an elementary proof that the sum of $p$-th powers of the first $n$ natural numbers can be expressed as a polynomial in $n$ of degree $p+1$. We also prove a novel identity involving Bernoulli numbers and use it to show symmetry of this polynomial.

math.HO↗