# Bernoulli & Betti numbers (of manifolds) and the prime 34511

The purpose of this question is to resolve a mystery surrounding the prime 34511 that has got me bogged down for a while now. If you only care about the number theory and not the motivation coming from topology, skip to the very last paragraph.

M. Kreck and D. Zagier have a project addressing the question which sequences of numbers arise as rational Betti numbers of manifolds. Here is a talk by Zagier on this project that you can watch online. I attended another talk by Kreck on the same project. As far as I am aware, no preprint has appeared yet, unfortunately.

Let me summarize some of their results that I find most interesting: consider (closed, connected, smooth, oriented) manifolds $$M$$ of dimension $$4m$$ whose (rational) Betti numbers $$b_i$$ all vanish except $$b_0 = b_{4m} = 1$$ and $$b_{2m}$$ which is required to be odd. If such a manifold exists, let $$\tilde{b}_{2m}$$ denote the smallest possible value of $$b_{2m}$$; if none exist, set $$\tilde{b}_{2m} = \infty$$. Then Kreck and Zagier prove:

• $$\tilde{b}_{2m} < \infty$$ iff $$m = 1$$ or $$m$$ is even and a sum of at most two powers of $$2$$.
• Even when $$\tilde{b}_{2m} < \infty$$, it is rarely $$1$$. The question when exactly it is $$1$$ has been studied by Z. Su in her thesis (U Indiana, advised by J. Davis) and later in several papers (some of them are joint work, with Fowler/Kennard). The keyword here is rational projectvie plane. Here is a talk on these results. There is no complete answer yet, but many things are known.
• If $$1 < b_{2m} < \infty$$, typically it is $$5$$. But sometimes it is much larger, for instance Kreck mentioned that $$\tilde{b}_{2^{24}} = 34511$$.

The prime number 34511 has, surprisingly, also appeared in my own work (joint with M. Krannich), in a similar but not identical context: in Remark 2.20 of this paper together with the earlier statements, it is mentioned that the smallest possible dimension $$4m = 8\ell$$ in which the smallest possible signature of a $$(2m-1)$$-connected $$4m$$-manifold is divisible by an odd prime is $$4\cdot 2678$$, and the odd prime is $$34511$$, again!

In both settings, Bernoulli numbers appear; more concretely, $$B_{2k}$$ for $$k = m$$ and $$k = m/2$$ if $$m$$ is even. Recall that $$B_{2k} = \frac{(-1)^{k+1}2(2k)!}{(2\pi)^{2k}}\zeta(2k)$$. So you might first think it is not too surprising the same prime 34511 pops up twice. But what is strange is that the indices that are relevant in each case are rather different: in the work by Kreck and Zagier, it is a power of $$2$$, in our computation it is $$2678 = 2 \cdot 13 \cdot 103$$. Note that to get $$34511$$, it is crucial that the gcd of $$(2^{k-1}-1)\text{num}\left(\frac{|B_k|}{2k}\right), k = 2\ell, 4\ell$$ is taken. One may also ask if $$\text{num}\left(\frac{|B_k|}{2k}\right)$$ for $$k = 2\ell, 4\ell$$ are always coprime and probably the answer is no, but the smallest counterexample is simply too big to be found. My (admittedly rather vague) question is whether 34511 is in some sense special, in particular in relation to Bernoulli numbers.

• The $p$-divisibility of the Bernoulli number $B_{2k}$ depends only on $2k \mod p-1$ (slightly simplified) so the reason $2^{23}$ shows up in one case and $2678$ shows up in the other should be because of the congruence $2^{23} \equiv 2678 \mod (34511-1)$. – Will Sawin Mar 31 at 15:35
• Similarly $p$ divides $2^{k-1}-1$ if and only if $k$ is congruent to $1$ modulo the order of $2$ mod $p$, which itself divides $p-1$. For $34511$ this is $595$. Thus once $k$ is a solution, anything congruent to $k$ mod $p-1$ is a solution. – Will Sawin Mar 31 at 15:45
• $-\tfrac{1}{3}9^9 B_{18}(1/3)/B_{18}=1871\cdot 34511$ – Carlo Beenakker Mar 31 at 16:54
• I don't know if this is how Carlo Beenakker did it, but one way to arrive at his identity is to search the OEIS for "34511 bernoulli"; this will take you to A096053. The OEIS will also tell you that 34511 divides $3^{17}-1$ (see A074477). – Timothy Chow Apr 1 at 13:46