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.
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
xn + yn = zn
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
x2 + y2 = z2 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 xn + yn = zn then
if n = pq,
(xq)p + (yq)p = (zq)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
p2 + 3q2
and Euler shows that, for any a, b if we put
p = a3 - 9ab2, q = 3(a2b - b3) then
p2 + 3q2 = (a2 + 3b2)3.
This is true but he then tries to show that, if p2 + 3q2 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 xn + yn = zn 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 xn + yn =
zn 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 B2 ,
B4 , ..., Bp-3 . The Bernoulli number Bi is defined by
x/(ex - 1) = Bi xi /i!
Kummer shows that all primes up to 37 are regular but 37 is not regular as 37
divides the numerator of B32 .
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, Furtwنngler, 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 xn + yn
= zn. 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 y2 = x3 + 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.
