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Archive-name: sci-math-faq/fermat
Last-modified: February 20, 1998
Version: 7.5

   
                             Fermat's Last Theorem
                                       
History of Fermat's Last Theorem

   Pierre de Fermat (1601-1665) was a lawyer and amateur mathematician.
   In about 1637, he annotated his copy (now lost) of Bachet's
   translation of Diophantus' Arithmetika with the following statement:
   
     Cubum autem in duos cubos, aut quadratoquadratum in duos
     quadratoquadratos, et generaliter nullam in infinitum ultra
     quadratum potestatem in duos ejusdem nominis fas est dividere:
     cujus rei demonstrationem mirabilem sane detexi. Hanc marginis
     exiguitas non caperet.
     
   In English, and using modern terminology, the paragraph above reads
   as:
   
     There are no positive integers such that x^n + y^n = z^n for n>2.
     I've found a remarkable proof of this fact, but there is not enough
     space in the margin [of the book] to write it.
     
   Fermat never published a proof of this statement. It became to be
   known as Fermat's Last Theorem (FLT) not because it was his last piece
   of work, but because it is the last remaining statement in the
   post-humous list of Fermat's works that needed to be proven or
   independently verified. All others have either been shown to be true
   or disproven long ago.
   
What is the current status of FLT?

   Theorem 1 [Fermat's Last Theorem] There are no positive integers x, y,
   z, and n > 2 such that x^n + y^n = z^n.
   
   Andrew Wiles, a researcher at Princeton, claims to have found a proof.
   The proof was presented in Cambridge, UK during a three day seminar to
   an audience which included some of the leading experts in the field.
   The proof was found to be wanting. In summer 1994, Prof. Wiles
   acknowledged that a gap existed. On October 25th, 1994, Prof. Andrew
   Wiles released two preprints, Modular elliptic curves and Fermat's
   Last Theorem, by Andrew Wiles, and Ring theoretic properties of
   certain Hecke algebras, by Richard Taylor and Andrew Wiles. The first
   one (long) announces a proof of, among other things, Fermat's Last
   Theorem, relying on the second one (short) for one crucial step.
   
   The argument described by Wiles in his Cambridge lectures had a
   serious gap, namely the construction of an Euler system. After trying
   unsuccessfully to repair that construction, Wiles went back to a
   different approach he had tried earlier but abandoned in favor of the
   Euler system idea. He was able to complete his proof, under the
   hypothesis that certain Hecke algebras are local complete
   intersections. This and the rest of the ideas described in Wiles'
   Cambridge lectures are written up in the first manuscript. Jointly,
   Taylor and Wiles establish the necessary property of the Hecke
   algebras in the second paper.
   
   The new approach turns out to be significantly simpler and shorter
   than the original one, because of the removal of the Euler system. (In
   fact, after seeing these manuscripts Faltings has apparently come up
   with a further significant simplification of that part of the
   argument.)
   
   The papers were published in the May 1995 issue of Annals of
   Mathematics. For single copies of the issues send e-mail to
   jlorder@jhunix.hcf.jhu.edu for further directions.
   
   In summary:
   
   Both manuscripts have been published. Thousands of people have a read
   them. About a hundred understand it very well. Faltings has simplified
   the argument already. Diamond has generalized it. People can read it.
   The immensely complicated geometry has mostly been replaced by simpler
   algebra. The proof is now generally accepted. There was a gap in this
   second proof as well, but it has been filled since October 1994.
   
Related Conjectures

   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 too many pages.
   
      References
      
   [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.
   Birkhduser Boston, Inc., 1982, ISBN 3-7643-3104-6
   
Did Fermat prove this theorem?

   No he did not. Fermat claimed to have found a proof of the theorem at
   an early stage in his career. Much later he spent time and effort
   proving the cases n=4 and n=5. Had he had a proof to his theorem
   earlier, there would have been no need for him to study specific
   cases.
   
   Fermat may have had one of the following ``proofs'' in mind when he
   wrote his famous comment.
   
     * Fermat discovered and applied the method of infinite descent,
       which, in particular can be used to prove FLT for n=4. This method
       can actually be used to prove a stronger statement than FLT for
       n=4, viz, x^4 + y^4 = z^2 has no non-trivial integer solutions. It
       is possible and even likely that he had an incorrect proof of FLT
       using this method when he wrote the famous ``theorem''.
     * He had a wrong proof in mind. The following proof, proposed first
       by Lame' was thought to be correct, until Liouville pointed out
       the flaw, and by Kummer which latter became and expert in the
       field. It is based on the incorrect assumption that prime
       decomposition is unique in all domains.
       The incorrect proof goes something like this:
       We only need to consider prime exponents (this is true). So
       consider x^p + y^p = z^p. Let r be a primitive p-th root of unity
       (complex number)
       Then the equation is the same as:
       (x + y)(x + ry)(x + r^2y)...(x + r^(p - 1)y) = z^p
       Now consider the ring of the form:
       a_1 + a_2 r + a_3 r^2 + ... + a_(p - 1) r^(p - 1)
       where each a_i is an integer
       Now if this ring is a unique factorization ring (UFR), then it is
       true that each of the above factors is relatively prime.
       From this it can be proven that each factor is a pth power from
       which FLT follows. This is usually done by considering two cases,
       the first where p divides none of x, y, z; the second where p
       divides some of x, y, z. For the first case, if x + yr = u*t^p,
       where u is a unit in Z[r] and t is in Z[r], it follows that x = y
       (mod p). Writing the original equation as x^p + (-z)^p = (-y)^p,
       it follows in a similar fashion that x = -z (mod p). Thus 2*x^p =
       x^p + y^p = z^p = -x^p (mod p) which implies 3*x^p = 0 (modp) and
       from there p divides one of x or 3|x. But p>3 and p does not
       divides x; contradiction. The second case is harder.
       The problem is that the above ring is not an UFR in general.
       
   Another argument for the belief that Fermat had no proof ---and,
   furthermore, that he knew that he had no proof--- is that the only
   place he ever mentioned the result was in that marginal comment in
   Bachet's Diophantus. If he really thought he had a proof, he would
   have announced the result publicly, or challenged some English
   mathematician to prove it. It is likely that he found the flaw in his
   own proof before he had a chance to announce the result, and never
   bothered to erase the marginal comment because it never occurred to
   him that anyone would see it there.
   
   Some other famous mathematicians have speculated on this question.
   Andre Weil, writes:
   
     Only on one ill-fated occasion did Fermat ever mention a curve of
     higher genus x^n + y^n = z^n, and then hardly remains any doubt
     that this was due to some misapprehension on his part [...] for a
     brief moment perhaps [...] he must have deluded himself into
     thinking he had the principle of a general proof.
     
   Winfried Scharlau and Hans Opolka report:
   
     Whether Fermat knew a proof or not has been the subject of many
     speculations. The truth seems obvious ... [Fermat's marginal note]
     was made at the time of his first letters concerning number theory
     [1637]... as far as we know he never repeated his general remark,
     but repeatedly made the statement for the cases n=3 and 4 and posed
     these cases as problems to his correspondents [...] he formulated
     the case n=3 in a letter to Carcavi in 1659 [...] All these facts
     indicate that Fermat quickly became aware of the incompleteness of
     the [general] ``proof" of 1637. Of course, there was no reason for
     a public retraction of his privately made conjecture.
     
   However it is important to keep in mind that Fermat's ``proof"
   predates the Publish or Perish period of scientific research in which
   we are still living.
   
      References
      
   From Fermat to Minkowski: lectures on the theory of numbers and its
   historical development. Winfried Scharlau, Hans Opolka. New York,
   Springer, 1985.
   
   Basic Number Theory. Andre Weil. Berlin, Springer, 1967
     _________________________________________________________________
   
-- 
Alex Lopez-Ortiz                                         alopez-o@unb.ca
http://www.cs.unb.ca/~alopez-o                       Assistant Professor	
Faculty of Computer Science                  University of New Brunswick

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