From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Fermat primes?
Date: 5 Apr 1996 19:12:31 GMT

In article <00002098+00007ad3@msn.com>, Dan Koppel <Mangoes88@msn.com> wrote:
>  In Fermat's last theorem it is sufficient to prove that the 
>equation cannot be solved if for any exponent n equal to a prime 
>number.  In the 19th century a German mathematician called Ernst 
>Kummer proved Fermat's theorem for a subset of the primes called the 
>"regular primes".  This proof took care of about 60% of the primes.  

That's very old data about primes _in a given range_ (through about 4000,
for example). Jensen showed that the set of irregular primes is infinite;
I am not up to date on work in this area but the last time I checked it
was still unknown whether the collection of regular primes had positive
density, or was even infinite.

>I am very curious to know how these "regular primes" are defined.  

The defining property is that  p  not divide the order of the class
group of  Q[ zeta ]  where  zeta^p=1. There are equivalent statements
involving smaller and/or more easily-defined numbers (e.g. class numbers
of the maximal real subfield; Bernoulli numerators, etc.) The list of
irregular primes begins  37, 59, 67, 101, ...

Kummer's result is very pretty and rather natural. The whole point of
regularity is that it guarantees that ideals which appear to be
p-th powers really are p-th powers, so that a unique-factorization sort
of proof goes through mutatis mutandis.

dave