I hope that this does not read like a flaming response. I have never seen
anything
that I could call "easy" in any aspect of modular forms, functions,
automorphic functions,
elliptic (and other algebraic) curves, or any of the other supporting struts
in the background
for Taniyama-Shimura or Wiles' work. What little I know I find generally
confusing and very
tough. Maybe if I pound the rocks 18 hours a day for 6 years I could get
closer to a real understanding,
but I have to settle for real time efforts (3 hours a day for the next 36
years...) The toughest math course
I ever took was a graduate seminar of Shimura's (long ago) on automorphic
forms.
There may well be an easy way to do this stuff but I do not think anybody
knows what it is yet. By all means look!
Sometimes the computer application is simpler than the number theory.
The easiest related derivation I know is showing how even values of the
Riemann zeta function are
related to coefficients of the cotangent series. Nobody even has a "nice"
formula for the Bernoulli constants yet.
(And they have been on the table for centuries.) (IMO, the LL test is nice
compared to finding Bernoulli numbers.)
The simplest pair of modular forms parametrizing an elliptic curve are the
Weierstrass P function and its derivative.
These are not easy functions to mess with but they are certainly the best
known.
Joth
----- Original Message -----
From: Geoffrey Faivre-Malloy <[EMAIL PROTECTED]>
To: <[EMAIL PROTECTED]>
Sent: Sunday, July 25, 1999 9:40 AM
Subject: Mersenne: Stepping out on a limb here
> I'm rather new to much of the theory behind all of this so have mercy :)
>
> I just finished reading Fermat's Last Theorem which is a fascinating book.
> This introduced me to the Taniyama-Shimura conjecture and subsequent
> theorem. I've noticed that there are algorithms based on Elliptic Curves.
> The Taniyama Shimura theorem says that you can directly map each Elliptic
> curve to it's Modular form.
>
> Recently, a new theorem was proved that lets you solve Elliptic equations
> easily with the Modular form - see
> http://www.academicpress.com/inscight/07091999/grapha.htm for details.
>
> So, based on all of this, would there be some way write a program that
used
> these details to factor faster?
>
> G-Man
>
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