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

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 $p$.

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

where $m\equiv n\cancel{\equiv }0(modp-1)$.

valid when $m\equiv n(mod(p-1)p^{\mathrm{\ell }})$ and $n\cancel{\equiv }0(modp-1)$, where $\mathrm{\ell }(\ge 0)$ is a fixed integer.

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

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

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

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

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

With $N_{2n}$ as in §24.10(iii)

valid for fixed integers $\mathrm{\ell }(\ge 1)$, and for all $n(\ge 1)$ such that $2n\cancel{\equiv }0$ $(modp-1)$ and $p^{\mathrm{\ell }}|2n$.

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