# §24.10 Arithmetic Properties

## §24.10(i) Von Staudt–Clausen Theorem

Here and elsewhere in §24.10 the symbol $p$ denotes a prime number.

 24.10.1 $B_{2n}+\sum_{(p-1)\mathbin{|}2n}\frac{1}{p}=\hbox{integer},$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $n$: integer and $p$: prime Permalink: http://dlmf.nist.gov/24.10.E1 Encodings: TeX, pMML, png See also: Annotations for 24.10(i), 24.10 and 24

where the summation is over all $p$ such that $p-1$ divides $2n$. The denominator of $B_{2n}$ is the product of all these primes $p$.

 24.10.2 $pB_{2n}\equiv p-1\pmod{p^{\ell+1}},$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\ell$: integer, $n$: integer and $p$: prime Referenced by: §24.10(i) Permalink: http://dlmf.nist.gov/24.10.E2 Encodings: TeX, pMML, png See also: Annotations for 24.10(i), 24.10 and 24

where $n\geq 2$, and $\ell(\geq 1)$ is an arbitrary integer such that $(p-1)p^{\ell}\mathbin{|}2n$. Here and elsewhere two rational numbers are congruent if the modulus divides the numerator of their difference.

## §24.10(ii) Kummer Congruences

 24.10.3 $\frac{B_{m}}{m}\equiv\frac{B_{n}}{n}\pmod{p},$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $m$: integer, $n$: integer and $p$: prime Permalink: http://dlmf.nist.gov/24.10.E3 Encodings: TeX, pMML, png See also: Annotations for 24.10(ii), 24.10 and 24

where $m\equiv n\not\equiv 0\pmod{p-1}$.

 24.10.4 $(1-p^{m-1})\frac{B_{m}}{m}\equiv(1-p^{n-1})\frac{B_{n}}{n}\pmod{p^{\ell+1}},$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\ell$: integer, $m$: integer, $n$: integer and $p$: prime Permalink: http://dlmf.nist.gov/24.10.E4 Encodings: TeX, pMML, png See also: Annotations for 24.10(ii), 24.10 and 24

valid when $m\equiv n\pmod{(p-1)p^{\ell}}$ and $n\not\equiv 0\pmod{p-1}$, where $\ell(\geq 0)$ is a fixed integer.

 24.10.5 $E_{n}\equiv E_{n+p-1}\pmod{p},$ ⓘ Symbols: $E_{\NVar{n}}$: Euler numbers, $n$: integer and $p$: prime Referenced by: §24.10(ii) Permalink: http://dlmf.nist.gov/24.10.E5 Encodings: TeX, pMML, png See also: Annotations for 24.10(ii), 24.10 and 24

where $p(>2)$ is a prime and $n\geq 2$.

 24.10.6 $E_{2n}\equiv E_{2n+w}\pmod{2^{\ell}},$ ⓘ Symbols: $E_{\NVar{n}}$: Euler numbers, $\ell$: integer and $n$: integer Referenced by: §24.10(ii) Permalink: http://dlmf.nist.gov/24.10.E6 Encodings: TeX, pMML, png See also: Annotations for 24.10(ii), 24.10 and 24

valid for fixed integers $\ell(\geq 0)$, and for all $n(\geq 0)$ and $w(\geq 0)$ such that $2^{\ell}\mathbin{|}w$.

## §24.10(iii) Voronoi’s Congruence

Let $B_{2n}=\ifrac{N_{2n}}{D_{2n}}$, with $N_{2n}$ and $D_{2n}$ relatively prime and $D_{2n}>0$. Then

 24.10.7 $(b^{2n}-1)N_{2n}\equiv{2nb^{2n-1}D_{2n}\sum_{k=1}^{M-1}k^{2n-1}\left\lfloor% \frac{kb}{M}\right\rfloor\pmod{M}},$

where $M(\geq 2)$ and $b$ are integers, with $b$ relatively prime to $M$.

For historical notes, generalizations, and applications, see Porubský (1998).

## §24.10(iv) Factors

With $N_{2n}$ as in §24.10(iii)

 24.10.8 $N_{2n}\equiv 0\pmod{p^{\ell}},$ ⓘ Symbols: $\ell$: integer, $n$: integer and $p$: prime Referenced by: §24.10(iv) Permalink: http://dlmf.nist.gov/24.10.E8 Encodings: TeX, pMML, png See also: Annotations for 24.10(iv), 24.10 and 24

valid for fixed integers $\ell(\geq 1)$, and for all $n(\geq 1)$ such that $2n\not\equiv 0$ $\pmod{p-1}$ and $p^{\ell}\mathbin{|}2n$.

 24.10.9 $E_{2n}\equiv\begin{cases}0\pmod{p^{\ell}}&\text{if }p\equiv 1\pmod{4},\\ 2\pmod{p^{\ell}}&\text{if }p\equiv 3\pmod{4},\end{cases}$ ⓘ Symbols: $E_{\NVar{n}}$: Euler numbers, $\ell$: integer, $n$: integer and $p$: prime Referenced by: §24.10(iv) Permalink: http://dlmf.nist.gov/24.10.E9 Encodings: TeX, pMML, png See also: Annotations for 24.10(iv), 24.10 and 24

valid for fixed integers $\ell(\geq 1)$ and for all $n(\geq 1)$ such that $(p-1)p^{\ell-1}\mathbin{|}2n$.