Can someone explain this theorem from von Staudt on denominators of Bernoulli numbers

Question

This is an extract from Paulo Ribenboim: 13 lectures on Fermats last theorem on page 105. "In 1845, von Staudt determined some factors of the numerator $N_{2k}$. Let $2k = k_1k_2$ with $gcd(k_1,k_2)= 1$ such that $p|k_2$ if and only if $p|D_{2k}$ then $k_1$|$N_{2k}$". Where $N_{2k}$ and $D_{2k}$ are the numerators and denominators of Bernoulli number $B_{2k}$.

So I've actually used the result of this theorem for some other proof, but looking back at it I find it is not true. For example when $2k=74$, then $2k=2\cdot37$. If we take $p=37$, we see that $37|k_2=37$ and so 37 must divide the denominator $D_{74}$ but $D_{74}=6$. I'm not sure what I'm missing here. Perhaps I have misinterpreted the theorem. Could someone clear this up for me?

Answer 1

I can't explain what Ribenboim was trying to write, because it is not correct. Perhaps he meant to use the Von-Staudt Clausen theorem. The Wikipedia article states

Specifically, if $n$ is a positive integer and we add $1/p$ to the Bernoulli number $B_{2n}$ for every prime $p$ such that $p − 1$ divides $2n$, we obtain an integer [...]

This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers $B_{2n}$ as the product of all primes $p$ such that $p − 1$ divides $2n$; consequently the denominators are square-free and divisible by $6$.

In your case of $2n=74$ the only divisors of $74$ are $2$ and $37$ and only the primes $p=2$ and $p=3$ are such that $p-1$ divides $74$. Hence $D_{74}=6.$

As user Armatowski commented, there is a result of Ramanujan that is close to what is stated by Ribenboim. This is from Bruce C. Berndt, Ramanujan's Notebooks, Part I, page 123.

Entry 19(ii). The numerator of $B_{2n}$ is divisble by the largest factor for $2n$ which is relatively prime to the denominator of $B_{2n}$.

$\quad$ Entry 19(ii) is contained in (18) of Ramanujan's paper [4] and is originally due to J. C. Adams. (See Uspensky and Heaslet's book [1, p.261].) In fact, in both Entry 19(ii) and (18) of [4], Ramanujan claims a stronger result, viz., the implied quotient is a prime number. However, this is false, for example, the numerator of $B_{22}$ is $854513=11\cdot131\cdot593.$