Congruence properties of prime sums and Bernoulli polynomials

TL;DR

This paper studies congruence properties of sums involving primes, focusing on sums of the form $S(p,r)=sum_{k=1}^{p-r} floor(k^p/p)$. The authors derive a main congruence expressed via Bernoulli polynomials $B_{p+1}(x)$: $S(p,r) ≡ (B_{p+1}(r)-B_{p+1}(0))/(p(p+1)) + ((r-1-p)(p-r))/(2p)$ modulo $p$. They then connect these sums to sums of odd powers and obtain a key Bernoulli-polynomial congruence in the case $2q+1=p$, namely $(B_{p+1}(p+1)+B_{p+1}(p)-2B_{p+1})/(p(p+1)) ≡ p^{p-1}$ (mod $p$), with explicit examples for $p=3,5,7$ demonstrating the predicted residues. The approach uses Fermat's little theorem, Faulhaber-type expansions, and Bernoulli identities to derive modular relations, and it highlights connections to Wilson- and Wolstenholme-type congruences and potential avenues for additional sum-rule congruences.

Abstract

In this article, we derive a congruence property of particular sum rules involving prime numbers. The resulting expression involves Bernoulli numbers and polynomials, for which we obtain, as a consequence, a general congruence relation as well.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof