# A conjecture on p-divisibility of Bernoulli numbers

Is anyone aware of the history of the following conjecture on the $p$-divisibility of (the numerators) of Bernoulli numbers?

CONJECTURE: For $p$ an odd prime, and $k$ even with $2 \leq k \leq p-3$, $p$ cannot divide both $B_k$ and $B_{p-k+1}$.

This conjecture is mentioned in Ohta's paper: Companion Forms and the structure of $p$-adic Hecke algebras. (see Example 3.3.4 on p. 25)

As far as I am aware, this is the only instance of the conjecture in literature. I believe I have a proof using companion forms and Ribet's proof of Herbrand-Ribet. I am currently writing it up and want to give proper references. Thanks

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You might try emailing Ken Ribet. He will almost certainly answer you, and he's tremendously knowledgeable. –  Pete L. Clark Apr 12 '10 at 17:12
I think that the paper of Frank Calegari on Eisentein primes, in Compositio, also discusses this conjecture. (This is a somewhat vague memory, so I apologize in advance if I'm wrong.) –  Emerton Apr 12 '10 at 17:37
And if you do email Ribet, you might want to mention this MO question, I think he has registered here already (see user 5131). –  Thomas Sauvaget Apr 12 '10 at 17:47
By the way, this would be a very nice result, considering its application to the structure of Hecke algebras and to Iwasawa theory. –  Olivier Apr 13 '10 at 8:30
@unramified: Have you finally heard from Ribet ? And have you written up your proof of Ohta's conjecture ? –  Chandan Singh Dalawat Mar 14 '12 at 9:10