Fermat's last theorem

Pierre de Fermat died in 1665. Today we think of Fermat as a number theorist, in fact as perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was in fact a lawyer and only an amateur mathematician. Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous article written as an appendix to a colleague's book.

You can see a statue of Fermat and his muse in his home town of Toulouse at THIS LINK

Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son, Samuel undertook the task of collecting Fermat's letters and other mathematical papers, comments written in books, etc. with the object of publishing his father's mathematical ideas. In this way the famous 'Last theorem' came to be published. It was found by Samuel written as a marginal note in his father's copy of Diophantus's Arithmetica.

Fermat's Last Theorem states that
$x^{n} + y^{n} = z^{n}$
has no non-zero integer solutions for $x, y$ and $z$ when $n > 2$. Fermat wrote
I have discovered a truly remarkable proof which this margin is too small to contain.
Fermat almost certainly wrote the marginal note around 1630, when he first studied Diophantus's Arithmetica. It may well be that Fermat realised that his remarkable proof was wrong, however, since all his other theorems were stated and restated in challenge problems that Fermat sent to other mathematicians. Although the special cases of $n = 3$ and $n = 4$ were issued as challenges (and Fermat did know how to prove these) the general theorem was never mentioned again by Fermat.

In fact in all the mathematical work left by Fermat there is only one proof. Fermat proves that the area of a right triangle cannot be a square. Clearly this means that a rational triangle cannot be a rational square. In symbols, there do not exist integers $x, y, z$ with
$x^{2} + y^{2} = z^{2}$ such that $xy/2$ is a square. From this it is easy to deduce the $n = 4$ case of Fermat's theorem.

It is worth noting that at this stage it remained to prove Fermat's Last Theorem for odd primes $n$ only. For if there were integers $x, y, z$ with $x^{n} + y^{n} = z^{n}$ then if $n = pq$,
$(x^{q})^{p} + (y^{q})^{p} = (z^{q})^{p}$.
Euler wrote to Goldbach on 4 August 1753 claiming he had a proof of Fermat's Theorem when $n = 3$. However his proof in Algebra (1770) contains a fallacy and it is far from easy to give an alternative proof of the statement which has the fallacious proof. There is an indirect way of mending the whole proof using arguments which appear in other proofs of Euler so perhaps it is not too unreasonable to attribute the $n = 3$ case to Euler.

Euler's mistake is an interesting one, one which was to have a bearing on later developments. He needed to find cubes of the form
$p^{2} + 3q^{2}$
and Euler shows that, for any $a, b$ if we put
$p = a^{3} - 9ab^{2}, q = 3(a^{2}b - b^{3})$ then
$p^{2} + 3q^{2} = (a^{2} + 3b^{2})^{3}.$
This is true but he then tries to show that, if $p^{2} + 3q^{2}$ is a cube then an $a$ and $b$ exist such that $p$ and $q$ are as above. His method is imaginative, calculating with numbers of the form $a + b√-3$. However numbers of this form do not behave in the same way as the integers, which Euler did not seem to appreciate.

The next major step forward was due to Sophie Germain. A special case says that if $n$ and $2n + 1$ are primes then $x^{n} + y^{n} = z^{n}$ implies that one of $x, y, z$ is divisible by $n$. Hence Fermat's Last Theorem splits into two cases.
Case 1: None of $x, y, z$ is divisible by $n$.
Case 2: One and only one of $x, y, z$ is divisible by $n$.
Sophie Germain proved Case 1 of Fermat's Last Theorem for all $n$ less than 100 and Legendre extended her methods to all numbers less than 197. At this stage Case 2 had not been proved for even $n = 5$ so it became clear that Case 2 was the one on which to concentrate. Now Case 2 for $n = 5$ itself splits into two. One of $x, y, z$ is even and one is divisible by 5. Case 2(i) is when the number divisible by 5 is even; Case 2(ii) is when the even number and the one divisible by 5 are distinct.

Case 2(i) was proved by Dirichlet and presented to the Paris Académie des Sciences in July 1825. Legendre was able to prove Case 2(ii) and the complete proof for $n = 5$ was published in September 1825. In fact Dirichlet was able to complete his own proof of the $n = 5$ case with an argument for Case 2(ii) which was an extension of his own argument for Case 2(i).

In 1832 Dirichlet published a proof of Fermat's Last Theorem for $n = 14$. Of course he had been attempting to prove the $n = 7$ case but had proved a weaker result. The $n = 7$ case was finally solved by Lamé in 1839. It showed why Dirichlet had so much difficulty, for although Dirichlet's $n = 14$ proof used similar (but computationally much harder) arguments to the earlier cases, Lamé had to introduce some completely new methods. Lamé's proof is exceedingly hard and makes it look as though progress with Fermat's Last Theorem to larger $n$ would be almost impossible without some radically new thinking.

The year 1847 is of major significance in the study of Fermat's Last Theorem. On 1 March of that year Lamé announced to the Paris Académie that he had proved Fermat's Last Theorem. He sketched a proof which involved factorizing $x^{n} + y^{n} = z^{n}$ into linear factors over the complex numbers. Lamé acknowledged that the idea was suggested to him by Liouville. However Liouville addressed the meeting after Lamé and suggested that the problem of this approach was that uniqueness of factorisation into primes was needed for these complex numbers and he doubted if it were true. Cauchy supported Lamé but, in rather typical fashion, pointed out that he had reported to the October 1847 meeting of the Académie an idea which he believed might prove Fermat's Last Theorem.

Much work was done in the following weeks in attempting to prove the uniqueness of factorization. Wantzel claimed to have proved it on 15 March but his argument
It is true for $n = 2, n = 3$ and $n = 4$ and one easily sees that the same argument applies for $n > 4$
was somewhat hopeful.
[Wantzel is correct about $n = 2$ (ordinary integers), $n = 3$ (the argument Euler got wrong) and $n = 4$ (which was proved by Gauss).]

On 24 May Liouville read a letter to the Académie which settled the arguments. The letter was from Kummer, enclosing an off-print of a 1844 paper which proved that uniqueness of factorization failed but could be 'recovered' by the introduction of ideal complex numbers which he had done in 1846. Kummer had used his new theory to find conditions under which a prime is regular and had proved Fermat's Last Theorem for regular primes. Kummer also said in his letter that he believed 37 failed his conditions.

By September 1847 Kummer sent to Dirichlet and the Berlin Academy a paper proving that a prime $p$ is regular (and so Fermat's Last Theorem is true for that prime) if $p$ does not divide the numerators of any of the Bernoulli numbers $B_{2} , B_{4} , ..., B_{p-3}$ . The Bernoulli number $B_{i}$ is defined by
$x/(e^{x} - 1) = \sum_{i=0}^{\infty} B_{i} x^{i} /i!$
Kummer shows that all primes up to 37 are regular but 37 is not regular as 37 divides the numerator of $B_{32}$ .

The only primes less than 100 which are not regular are 37, 59 and 67. More powerful techniques were used to prove Fermat's Last Theorem for these numbers. This work was done and continued to larger numbers by Kummer, Mirimanoff, Wieferich, Furtwängler, Vandiver and others. Although it was expected that the number of regular primes would be infinite even this defied proof. In 1915 Jensen proved that the number of irregular primes is infinite.

Despite large prizes being offered for a solution, Fermat's Last Theorem remained unsolved. It has the dubious distinction of being the theorem with the largest number of published false proofs. For example over 1000 false proofs were published between 1908 and 1912. The only positive progress seemed to be computing results which merely showed that any counter-example would be very large. Using techniques based on Kummer's work, Fermat's Last Theorem was proved true, with the help of computers, for $n$ up to 4,000,000 by 1993.

In 1983 a major contribution was made by Gerd Faltings who proved that for every $n > 2$ there are at most a finite number of coprime integers $x, y, z$ with $x^{n} + y^{n} = z^{n}$. This was a major step but a proof that the finite number was 0 in all cases did not seem likely to follow by extending Faltings' arguments.

The final chapter in the story began in 1955, although at this stage the work was not thought of as connected with Fermat's Last Theorem. Yutaka Taniyama asked some questions about elliptic curves, i.e. curves of the form $y^{2} = x^{3} + ax + b$ for constants $a$ and $b$. Further work by Weil and Shimura produced a conjecture, now known as the Shimura-Taniyama-Weil Conjecture. In 1986 the connection was made between the Shimura-Taniyama- Weil Conjecture and Fermat's Last Theorem by Frey at Saarbrücken showing that Fermat's Last Theorem was far from being some unimportant curiosity in number theory but was in fact related to fundamental properties of space.

Further work by other mathematicians showed that a counter-example to Fermat's Last Theorem would provide a counter -example to the Shimura-Taniyama-Weil Conjecture. The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA. Wiles gave a series of three lectures at the Isaac Newton Institute in Cambridge, England the first on Monday 21 June, the second on Tuesday 22 June. In the final lecture on Wednesday 23 June 1993 at around 10.30 in the morning Wiles announced his proof of Fermat's Last Theorem as a corollary to his main results. Having written the theorem on the blackboard he said I will stop here and sat down. In fact Wiles had proved the Shimura-Taniyama-Weil Conjecture for a class of examples, including those necessary to prove Fermat's Last Theorem.

This, however, is not the end of the story. On 4 December 1993 Andrew Wiles made a statement in view of the speculation. He said that during the reviewing process a number of problems had emerged, most of which had been resolved. However one problem remains and Wiles essentially withdrew his claim to have a proof. He states
The key reduction of (most cases of) the Taniyama-Shimura conjecture to the calculation of the Selmer group is correct. However the final calculation of a precise upper bound for the Selmer group in the semisquare case (of the symmetric square representation associated to a modular form) is not yet complete as it stands. I believe that I will be able to finish this in the near future using the ideas explained in my Cambridge lectures.
In March 1994 Faltings, writing in Scientific American, said
If it were easy, he would have solved it by now. Strictly speaking, it was not a proof when it was announced.
Weil, also in Scientific American, wrote
I believe he has had some good ideas in trying to construct the proof but the proof is not there. To some extent, proving Fermat's Theorem is like climbing Everest. If a man wants to climb Everest and falls short of it by 100 yards, he has not climbed Everest.
In fact, from the beginning of 1994, Wiles began to collaborate with Richard Taylor in an attempt to fill the holes in the proof. However they decided that one of the key steps in the proof, using methods due to Flach, could not be made to work. They tried a new approach with a similar lack of success. In August 1994 Wiles addressed the International Congress of Mathematicians but was no nearer to solving the difficulties.

Taylor suggested a last attempt to extend Flach's method in the way necessary and Wiles, although convinced it would not work, agreed mainly to enable him to convince Taylor that it could never work. Wiles worked on it for about two weeks, then suddenly inspiration struck.
In a flash I saw that the thing that stopped it [the extension of Flach's method] working was something that would make another method I had tried previously work.
On 6 October Wiles sent the new proof to three colleagues including Faltings. All liked the new proof which was essentially simpler than the earlier one. Faltings sent a simplification of part of the proof.

No proof of the complexity of this can easily be guaranteed to be correct, so a very small doubt will remain for some time. However when Taylor lectured at the British Mathematical Colloquium in Edinburgh in April 1995 he gave the impression that no real doubts remained over Fermat's Last Theorem.

References (show)

1. A D Aczel, Fermat's last theorem : Unlocking the secret of an ancient mathematical problem (New York, 1996).
2. D A Cox, Introduction to Fermat's last theorem, Amer. Math. Monthly 101 (1) (1994), 3-14.
3. H M Edwards, Fermat's last theorem : A genetic introduction to algebraic number theory (New York, 1996).
4. H M Edwards, The background of Kummer's proof of Fermat's last theorem for regular primes, Arch. History Exact Sci. 14 (3) (1975), 219-236.
5. C Goldstein, Le theoreme de Fermat, La recherche 263 (1994), 268-275.
6. D R Heath-Brown, The first case of Fermat's last theorem, Math. Intelligencer 7 (4) (1985), 40-47; 55.
7. R de Castro Korgi, The proof of Fermat's last theorem has been announced in Cambridge, England (Spanish), Lect. Mat. 14 (1-3) (1993), 129-135.
8. F Nemenzo, Fermat's last theorem : a mathematical journey, Matimyás Mat. 17 (2) (1994), 1-11.
9. A van der Poorten, Notes on Fermat's last theorem (New York, 1996).
10. A van der Poorten, Remarks on Fermat's last theorem, Austral. Math. Soc. Gaz. 21 (5) (1994), 150-159.
11. P Ribenboim, 13 lectures on Fermat's last theorem (New York, 1979).
12. P Ribenboim, Kummer's ideas on Fermat's last theorem, Enseign. Math. (2) 29 (1-2) (1983), 165-177.
13. P Ribenboim, Fermat's last theorem, before June 23, 1993, in Number theory (Providence, RI, 1995), 279-294.
14. P Ribenboim, The history of Fermat's last theorem (Portuguese), Bol. Soc. Paran. Mat. (2) 5 (1) (1984), 14-32.
15. P Ribenboim, Recent results on Fermat's last theorem, Canad. Math. Bull. 20 (2) (1977), 229-242.
16. R Schoof, Fermat's last theorem, in Jahrbuch überblicke Mathematik (Braunschweig, 1995), 193-211.
17. S L Singh, Fermat's last Theorem (London 1997)
18. S Wagon, Fermat's last theorem, Math. Intelligencer 8 (1) (1986), 59-61.