From: ray steiner 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 Sent via Deja.com http://www.deja.com/