# A general family of congruences for Bernoulli numbers

## TL;DR

This paper introduces a broad family of congruences for Bernoulli numbers, extending classical results like von Staudt--Clausen and Kummer's congruence, applicable to polynomial indices of primes.

## Contribution

It generalizes existing congruences for Bernoulli numbers by establishing a unified framework for polynomial-indexed congruences modulo prime powers.

## Key findings

- Established a new family of congruences for Bernoulli numbers
- Unified and extended classical results like von Staudt--Clausen and Kummer's congruence
- Applicable to polynomial functions of primes as indices

## Abstract

We prove a general family of congruences for Bernoulli numbers whose index is a polynomial function of a prime, modulo a power of that prime. Our family generalizes many known results, including the von Staudt--Clausen theorem and Kummer's congruence.