Kummer's congruence

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 $k\ge 2$ 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 $k,k^{\prime }$ with $\text{min}(k,k^{\prime })\ge e+1$ and $(p-1)\nmid k$ and $k\equiv k^{\prime }(\text{mod}\phi (p^e))$, 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.