# Power-Sum Denominators

## TL;DR

This paper investigates the denominators of power sum polynomials and Bernoulli polynomials, revealing a prime factorization pattern related to base-p digit sums, with implications for understanding their arithmetic properties.

## Contribution

It establishes a new prime factorization formula for the denominators of power sum polynomials and Bernoulli polynomials based on digit sum conditions in various bases.

## Key findings

- Denominator equals (n+1) times the squarefree product of primes with specific digit sum properties.
- Provides a new formula for denominators of Bernoulli polynomials.
- Highlights a connection between prime factors and base-p digit sums.

## Abstract

The power sum $1^n + 2^n + \cdots + x^n$ has been of interest to mathematicians since classical times. Johann Faulhaber, Jacob Bernoulli, and others who followed expressed power sums as polynomials in $x$ of degree $n+1$ with rational coefficients. Here we consider the denominators of these polynomials, and prove some of their properties. A remarkable one is that such a denominator equals $n+1$ times the squarefree product of certain primes $p$ obeying the condition that the sum of the base-$p$ digits of $n+1$ is at least $p$. As an application, we derive a squarefree product formula for the denominators of the Bernoulli polynomials.