steve@anasazi.UUCP (Steve Villee) (09/04/85)
[in case someone eats this line]
Fermat's conjecture has always been a favorite of mine. If others
on the net share this interest, maybe we can get some discussion going.
(Maybe net.math.fermat will be created!) Here are a few topics.
(1) The so-called Case I attempts to demonstrate that x^p + y^p = z^p
is impossible for prime exponent p unless one of x, y, or z is divisible
by p. This demonstration is usually quite independent of the demonstration
for Case II. It seems that for many primes that are 5 modulo 6, this can be
shown simply by an exhaustive check of the equation modulo p^2. In fact,
the first 5 mod 6 prime p for which there is a Case I solution modulo p^2
is 59, which just happens to be the first "irregular" 5 mod 6 prime!
Is this just a coincidence? Are there infinitely many 5 mod 6 primes p
for which there is no Case I solution modulo p^2? I did a computer search
along these lines once, and found that the proportion of 5 mod 6 primes
that had a Case I solution modulo p^2 appeared to increase gradually,
and some of them (e.g. 83) were regular.
(2) The class number of Q(e^(2*pi*i/p)) is typically factored as h =
h1 * h2, where h2 is the class number of the real subfield Q(cos(2*pi/p)).
Has anyone ever actually calculated a value of h2, except when the value
is 1? Has anyone ever proved or refuted the conjecture that h2 is never
divisible by p? If not, has anyone ever shown that h2 >= p for some p?
It's been a few years since I followed the journals actively, so there
may well be some new results.
(3) Has anyone ever found an elementary proof of the class number formula,
or at least for certain cases? I know about B. A. Venkov's proof for
imaginary quadratic fields whose discriminant is -3 modulo 8, but if
anoyone has an English translation and/or a simplification of this proof,
I'd be very interested to hear about it.
--- Steve Villee (ihnp4!mot!anasazi!steve)
International Anasazi, Inc.
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Phoenix, Arizona 85034
(602) 275-0302