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sci.math FAQ: What if Wiles is wrong?

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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.


   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


   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,

       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

       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.


   [1] 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

    Tue Apr 04 17:26:57 EDT 1995

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