Bernoulli numbers and arithmetic progressions

[John Conway]

On Thu, 5 Feb 2004, cino hilliard quoted or wrote:
> >
> >all values yielding 37 are of the form: 574+666*k, k=0,1,2,3,4,...
> >and form thus an arithmetic progression with step 666=18*37=((37-1)/2)*37;
> >the two values yielding 67 are at distance 2211=((67-1)/2)*67
> >
> >Conjecture: all indices yielding a given prime p form an arithmetic
> >progression of step ((p-1)/2)*p.
> >
> >If yes, the only important data would be the beginning of the
> >arithmetic progression for a given prime.
> >
> >By the way, in order to check this conjecture experimentally in
> >a few more cases it is enough to compute the corresponding
> >p-valuations. This can be done using (for instance) a recursive
> >formula for Bernoulli numbers and working in the finite ring
> >Z/(p^a Z) with a>0 big enough (in any case, a such that p^a>2n
> >should work). This trick should allow to push computations quite a lot
> >further.

    What is the conjecture being spoken of here?  It sounds as though it's
some kind of congruence involving Bernoulli numbers, in which case I'd 
like to have a shot at it.
     John Conway