ArchiveName: scimathfaq/FLT/Wrong
Lastmodified: December 8, 1994 Version: 6.2 See reader questions & answers on this topic!  Help others by sharing your knowledge 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 nonzero, 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 viceversa] 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 nonsingular algebraic varieties . Vojta's conjecture also implies Mordell's theorem [already known to be true]. There are also a number of intertwined 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 TaniyamaWeilShmimura conjecture. There is a very important and well known conjecture known as the TaniyamaWeilShimura 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 TateShafarevich 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 nontrivial, that infinitedescent arguments do not work; that in some sense 'Sha blocks the descent'. Somewhat more technically, Sha is an obstruction to the localglobal principle [e.g. the HasseMinkowski 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) 825835. This synopsis is quite brief. A full survey would run to many pages. References [1] J.P.Butler, R.E.Crandall,&R.W.Sompolski, Irregular Primes to One Million. Math. Comp., 59 (October 1992) pp. 717722. 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 3764331046 _________________________________________________________________ alopezo@barrow.uwaterloo.ca Tue Apr 04 17:26:57 EDT 1995 User Contributions:
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