Let $B_n$ be $n$-th Bernoulli number. We will write $p \mid B_n$ if prime $p$ divides the numerator of $B_n$.
Now $p = 583$, we can show $p \mid B_{90}$ and $p \mid B_{92}$ (I found it using computer). So, there exists two consecutive non-zero Bernoulli numbers divisible by same prime.
Question : What is the maximum length of a sequence of consecutive non-zero Bernoulli numbers divisible by a sufficiently large prime $p$?
I have no idea how to progress. I'm looking for some ideas or some references. Thanks in advance.