Kummer's congruence

Manuel Eberl 📧

March 20, 2024

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.


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.


BSD License


Session Kummer_Congruence