From: ray steiner <steiner@math.bgsu.edu>
Subject: Class numbers of cyclotomic fields and Catalan's equation
Date: Thu, 21 Dec 2000 22:10:11 GMT
Newsgroups: sci.math
Summary: [missing]

Greetings, all!

     Here is a question about cyclotomic fields. Unfortunately,
I haven't been able to find a good answer to it in the literature.
Let p, q be odd primes with q > p. Let h_p^- be the class number
of the cyclotomic field Q(zeta_p). Is there a reasonably
fast algorithm(say one that takes O(p*log p) operations or so)
for determining if q divides h_p^-?
If one searches the literature one finds
1). Many ways of answering this question if q = p.

2). Many fine algorithms for computing h_p^-.
For example, Lehmer and Masley, Fung et al, Jha, Shokrollahi
and Louboutin have all given algorithms for this. Unfortunately,
the best of these seems to have a running time of about
O(p^1.5*log(p)) operations, which is too slow for the range
of p I would like to check(10^7 < p < 3.31*10^12).

3). Modular methods in the papers of Jha and Shokrollahi
for answering my question if q = 1(mod p-1). Is there
a similar method for general q?

This question is of great importance in searching for
prime pairs p,q for which Catalan's equation x^p-y^q=1
holds(i.e., Catalan pairs). If q doesn't divide h_p^-
then p, q csnnot be a Catalan pair.(This is Bugeaud's
big result from last year.)

One last question: If p, q is a Catalan pair, can
we have q = 1(mod p-1)?
Regards,
Ray Steiner

--
steiner@bgnet.bgsu.edu



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