### Abstract

This entry provides proofs for two important congruences involving Bernoulli numbers.
The proofs follow Cohen's textbook *Number Theory Volume II: Analytic and Modern Tools*.
In the following we write $\mathcal{B}_k = N_k / D_k$ for the $k$-th Bernoulli number (with $\text{gcd}(N_k, D_k) = 1$).

The first result that I showed is *Voronoi's congruence*, which states that for any even integer
$k\geq 2$ and all positive coprime integers $a$, $n$ we have:

Building upon this, I then derive *Kummer's congruence*. In its common form, it states that
for a prime $p$ and even integers $k,k'$ with $\text{min}(k,k')\geq e+1$ and $(p - 1) \nmid k$ and
$k \equiv k'\ (\text{mod}\ \varphi(p^e))$, we have:

The version proved in my entry is slightly more general than this.

One application of these congruences is to prove that there are infinitely many irregular primes, which I formalised as well.

### License

### Topics

### Session Kummer_Congruence

- Kummer_Library
- Rat_Congruence
- Voronoi_Congruence
- Kummer_Congruence
- Regular_Primes
- Irregular_Primes_Infinite