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Fermat's last theorem

For more than three centuries, Fermat's last theorem fascinated many mathematicians, and even after it was finally proved by Andrew Wiles in 1994, continues to exert great fascination. Pierre de Fermat stated the theorem in the margin of a page in Bachet's edition of Diophantus's complete works,[1] but the proof was not found in any of Fermat's extant papers. For over 300 years, the theorem was in fact a hypothesis or conjecture. Nevertheless there are a very few texts written prior to the 1990s which refer to it as "the Fermat conjecture."[2] And because Wiles rather than Fermat gave the first published and verified proof, some have even suggested Fermat's last theorem should be called Wiles' theorem (or Wiles's theorem) instead.[3] Also, there is reason to believe that it was in fact not the last theorem Fermat ever stated.

Theorem (Fermat). Given an integer ${\displaystyle \scriptstyle n\,>\,2\,}$, the equation ${\displaystyle \scriptstyle x^{n}+y^{n}\,=\,z^{n}\,}$ has no solutions using only positive integers. (Cf. A019590)

It must be remarked that for ${\displaystyle \scriptstyle n\,=\,2\,}$ the solutions are of course the Pythagorean triples as per the Pythagorean theorem.[4] (See A046083, A046084 and A009000).

In the famous margin, Fermat wrote that he had found a wonderful proof but the margin was too small to contain it. If he had in fact found such a proof, it would probably be short enough to quote or paraphrase here. Andrew Wiles' proof in Annals of Mathematics takes over a hundred pages in a 1995 issue.[5] Wiles' proof involves showing "that all semistable elliptic curves over ${\displaystyle \scriptstyle \mathbb {Q} \,}$ are indeed modular."[6] This only hints at that since the proof draws on so many modern mathematical discoveries, many doubt that Fermat actually came up with a valid proof for all cases. Fermat did prove the case of biquadrates (${\displaystyle \scriptstyle n\,=\,4\,}$,)[7] and Gauss proved it for cubes (${\displaystyle \scriptstyle n\,=\,3\,}$.)[8] Andrew Wiles had first presented his proof in 1993, but it was found to contain an error "involving the Kolyvagin-Flach method."[9] It took Wiles a year to fix the problem. He fixed it using the horizontal Iwasawa theory approach[10]which he had rejected three years earlier.[11]

In between Gauss and Wiles, many other mathematicians attempted to prove the theorem or specific cases thereof. Both Wieferich and Sophie Germain studied cases of ${\displaystyle \scriptstyle n\,}$ being coprime to ${\displaystyle \scriptstyle x\,}$, ${\displaystyle \scriptstyle y\,}$ and ${\displaystyle \scriptstyle z\,}$ with kinds of prime numbers that are now named after them, i.e. the Wieferich primes (Cf. A001220) and the Sophie Germain primes (Cf. A005384).[12]

"For a time in the 19th century," mathematicians tried "to imagine the equation ${\displaystyle \scriptstyle x^{n}+y^{n}\,=\,z^{n}\,}$ as being over the complex numbers, and to use the complex ${\displaystyle \scriptstyle n\,}$th root of unity ${\displaystyle \scriptstyle \zeta \,=\,e^{\frac {2\pi i}{n}}\,}$ to obtain the factorization (valid for odd ${\displaystyle \scriptstyle n\,}$) ${\displaystyle \scriptstyle x^{n}+y^{n}\,=\,(x+y)(x+\zeta y)\cdots (x+\zeta ^{n-1}y).\,}$" This failed because the ring ${\displaystyle \scriptstyle \mathbb {Z} [\zeta ]\,}$ of polynomials in ${\displaystyle \scriptstyle \zeta \,}$ is not a unique factorization domain.[13] The first mathematician to publicly announce an effort along such lines was Gabriel Lamé, who, at a meeting of the Paris Academy in 1847, declared he had proven the theorem thanks to an idea he got from Joseph Liouville; Liouville then came up to the podium and pointed out the lack of unique factorization.[14]

In the Star Trek: The Next Generation episode "The Royale," Captain Picard (Patrick Stewart) famously says that Fermat's last theorem has remained unsolved for 800 years. Besides nits as to when Fermat stated the theorem and when the episode takes place,[15] Picard's line is flatly contradicted by the reality of Wiles proving the theorem only a few years after the episode first aired. The show's continuity workers tried to spin this to mean that after Wiles others (Picard included) searched for other proofs of the theorem.[16] Even the writers tried to backpedal on Picard's statement since "the current Wiles-Taylor proof has been declared valid": in the Star Trek: Deep Space Nine episode "Facets," Jadzia Dax tells Tobin Dax[17] that his proof of the theorem is the most original since the one by Wiles.[18] After stating the problem is unsolved after centuries, Picard says that "it puts things in perspective. In our arrogance we feel we are so advanced, and yet we cannot unravel a simple knot tied by a part-time French mathematician working alone, without a computer."[19] Another explanation could be that Picard does not consider the Wiles proof to be valid. It appears that people will continue to look for that proof just too long for a margin but short enough for the average person to digest and comprehend long after Picard's last attempt.

Notes

1. P. Ribenboim, 13 Lectures on Fermat's Last Theorem. New York: Springer-Verlag (1979): p. 1.
2. Such as: Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach Mineola, New York: Dover Publications (1969, reprinted 2007): p. 139
3. Bolker again, in the preface to the Dover reprint, p. x
4. Alf van der Poorten. Notes on Fermat's Last Theorem New York: John Wiley & Sons, Inc. 1996, p. vii.
5. Fred Diamond and John Im, "Modular forms and modular curves" Seminar on Fermat's Last Theorem V. Kumar Murty, editor. CMS Conference Proceedings Volume 17. Providence, Rhode Island: American Mathematical Society for the Canadian Mathematical Society (1995) p. 133 [Wil2].
6. V. Kumar Murty, "Modular Elliptic Curves" Seminar on Fermat's Last Theorem V. Kumar Murty, editor. CMS Conference Proceedings Volume 17. Providence, Rhode Island: American Mathematical Society for the Canadian Mathematical Society (1995): p. 1.
7. Ribenboim (1979): p. 2.
8. Ribenboim (1979): p. 3.
9. Simon Singh, Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker and Company (1997) p. 257.
10. Aczel ibid. p. 133.
11. Aczel ibid. p. 132.
12. Eric W. Weisstein, Fermat's Last Theorem, from MathWorld — A Wolfram Web Resource..
13. Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem 3rd Ed. Natick, Massachusettts, A K Peters 2002 p. xii.
14. Amir D. Aczel, Fermat's Last Theorem New York, London: Four Walls Eight Windows (1996): p. 66.
15. Phil Farrand takes "last theorem" literally, and since Fermat died in 1665 and another episode establishes the story's present Earth calendar year as 2365, Picard should have said 700 years rather than 800. P. Farrand, The Nitpicker's Guide for Next Generation Trekkers Volume II. New York: Dell Publishing (1995): p. 178.
16. M. Okuda & D. Okuda, Star Trek Chronology: The History of the Future Simon & Schuster Inc. Pocket Books (1996) Chapter 2.0, The Twentieth Century, 1993, p. 22. It could be argued that Picard's miscalculation by a whole century casts doubt on his abilities in higher arithmetic.
17. The exact nature of the relationship between these two characters is too complicated to explain here.
18. P. Farrand The Nitpicker's Guide for Deep Space Nine Trekkers Dell Publishing, New York (1996) p. 283.
19. The episode was written by Keith Mills.