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Archive-Name: sci-math-faq/FLT/Wrong
Last-modified: December 8, 1994
Version: 6.2

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```If not, then what?

FLT is usually broken into 2 cases. The first case assumes (abc,n) = 1
. The second case is the general case.

WHAT HAS BEEN PROVED

First Case.

It has been proven true up to 7.568*10^(17) by the work of Wagstaff
&Tanner, Granville &Monagan, and Coppersmith. They all used extensions
of the Wiefrich criteria and improved upon work performed by Gunderson
and Shanks &Williams.

The first case has been proven to be true for an infinite number of
exponents by Adelman, Frey, et. al. using a generalization of the
Sophie Germain criterion

Second Case:

It has been proven true up to n = 150,000 by Tanner &Wagstaff. The
work used new techniques for computing Bernoulli numbers mod p and
improved upon work of Vandiver. The work involved computing the
irregular primes up to 150,000. FLT is true for all regular primes by
a theorem of Kummer. In the case of irregular primes, some additional
computations are needed. More recently, Fermat's Last Theorem has been
proved true up to exponent 4,000,000 in the general case. The method
used was essentially that of Wagstaff: enumerating and eliminating
irregular primes by Bernoulli number computations. The computations
were performed on a set of NeXT computers by Richard Crandall et al.

Since the genus of the curve a^n + b^n = 1 , is greater than or equal
to 2 for n > 3 , it follows from Mordell's theorem [proved by
Faltings], that for any given n , there are at most a finite number of
solutions.

CONJECTURES

There are many open conjectures that imply FLT. These conjectures come
from different directions, but can be basically broken into several
classes: (and there are interrelationships between the classes)
1. Conjectures arising from Diophantine approximation theory such as
the ABC conjecture, the Szpiro conjecture, the Hall conjecture,
etc.

For an excellent survey article on these subjects see the article
by Serge Lang in the Bulletin of the AMS, July 1990 entitled ``Old
and new conjectured diophantine inequalities".

Masser and Osterle formulated the following known as the ABC
conjecture:

Given epsilon > 0 , there exists a number C(epsilon) such that for
any set of non-zero, relatively prime integers a,b,c such that a +
b = c we have max (|a|, |b|, |c|) <= C(epsilon) N(abc)^(1 +
epsilon) where N(x) is the product of the distinct primes dividing
x .

It is easy to see that it implies FLT asymptotically. The
conjecture was motivated by a theorem, due to Mason that
essentially says the ABC conjecture is true for polynomials.

The ABC conjecture also implies Szpiro's conjecture [and
vice-versa] and Hall's conjecture. These results are all generally
believed to be true.

There is a generalization of the ABC conjecture [by Vojta] which
is too technical to discuss but involves heights of points on
non-singular algebraic varieties . Vojta's conjecture also implies
Mordell's theorem [already known to be true]. There are also a
number of inter-twined conjectures involving heights on elliptic
curves that are related to much of this stuff. For a more complete
discussion, see Lang's article.

2. Conjectures arising from the study of elliptic curves and modular
forms. - The Taniyama-Weil-Shmimura conjecture.

There is a very important and well known conjecture known as the
Taniyama-Weil-Shimura conjecture that concerns elliptic curves.
This conjecture has been shown by the work of Frey, Serre, Ribet,
et. al. to imply FLT uniformly, not just asymptotically as with
the ABC conj.

The conjecture basically states that all elliptic curves can be
parameterized in terms of modular forms.

There is new work on the arithmetic of elliptic curves. Sha, the
Tate-Shafarevich group on elliptic curves of rank 0 or 1. By the
way an interesting aspect of this work is that there is a close
connection between Sha, and some of the classical work on FLT. For
example, there is a classical proof that uses infinite descent to
prove FLT for n = 4 . It can be shown that there is an elliptic
curve associated with FLT and that for n = 4 , Sha is trivial. It
can also be shown that in the cases where Sha is non-trivial, that
infinite-descent arguments do not work; that in some sense 'Sha
blocks the descent'. Somewhat more technically, Sha is an
obstruction to the local-global principle [e.g. the
Hasse-Minkowski theorem].

3. Conjectures arising from some conjectured inequalities involving
Chern classes and some other deep results/conjectures in
arithmetic algebraic geometry.

This results are quite deep. Contact Barry Mazur [or Serre, or
Faltings, or Ribet, or ...]. Actually the set of people who DO
understand this stuff is fairly small.

The diophantine and elliptic curve conjectures all involve deep
properties of integers. Until these conjecture were tied to FLT,
FLT had been regarded by most mathematicians as an isolated
problem; a curiosity. Now it can be seen that it follows from some
deep and fundamental properties of the integers. [not yet proven
but generally believed].

A related conjecture from Euler

x^n + y^n + z^n = c^n has no solution if n is >= 4

Noam Elkies gave a counterexample, namely 2682440^4 + 15365639^4 +
18796760^4 = 20615673^4 . Subsequently, Roger Frye found the
absolutely smallest solution by (more or less) brute force: it is
95800^4 + 217519^4 + 414560^4 = 422481^4 . "Several years", Math.
Comp. 51 (1988) 825-835.

This synopsis is quite brief. A full survey would run to many pages.

References

 J.P.Butler, R.E.Crandall,&R.W.Sompolski, Irregular Primes to One
Million. Math. Comp., 59 (October 1992) pp. 717-722.

Fermat's Last Theorem, A Genetic Introduction to Algebraic Number
Theory. H.M. Edwards. Springer Verlag, New York, 1977.

Thirteen Lectures on Fermat's Last Theorem. P. Ribenboim. Springer
Verlag, New York, 1979.

Number Theory Related to Fermat's Last Theorem. Neal Koblitz, editor.
Birkh<E4>user Boston, Inc., 1982, ISBN 3-7643-3104-6

_________________________________________________________________

alopez-o@barrow.uwaterloo.ca
Tue Apr 04 17:26:57 EDT 1995

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