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 $k\geq 2$ and all positive coprime integers $a$, $n$ we have:

$$(a^k - 1) N_k \equiv k a^{k-1} D_k \sum_{m=1}^{n-1} m^{k-1} \left\lfloor\frac{ma}{n}\right\rfloor \hspace{3mm}(\text{mod}\ n)$$

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:

$$\frac{\mathcal{B}_k}{k} \equiv \frac{\mathcal{B}_{k'}}{k'}\hspace{3mm}(\text{mod}\ p^e)$$

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.