From wstein@ucsd.edu Thu Feb 16 17:05:12 2006
From: William Stein <wstein@ucsd.edu>
Organization: UC San Diego
To: Barry Smith <barryrsmith@gmail.com>
Subject: Re: bernfrac
Date: Thu, 16 Feb 2006 17:05:12 -0800
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On Thursday 16 February 2006 16:14, you wrote:
> I have a couple of results whereby I can
> calculate class numbers of cyclic cubic fields of prime conductor p
> for primes p of the form p =3D (a^2 + 27)/4 and of the form p =3D (1 +
> 27*b^2)/4 if I can compute a couple of Bernoulli numbers. =A0In
> particular, in the first case, the class number h is characterized by
> the congruence
>
> h \equiv -3/2 B_{r}/r B_{2*r}/(2*r) mod p where r is (p-1)/3
>
> and in the second case,
>
> h \equiv -1/18 B_{r}/r B_{2*r}/(2*r) mod p where r is (p-1)/3.
>
> I have been toying with the idea of making a big table of class
> numbers, but I am not very tech savvy and I have a lot to work on as
> it is.

It sounds like we should make a table of class numbers of
all cyclic cubic fields of prime conductor p for all primes
p < 10^5 of the special forms you list above.=20

William