Fermat's Last Theorem states that

xn + yn = zn

has no non-zero integer solutions for x, y and
z when n > 2
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 Bn is
defined by

x/(ex - 1) = Bn xn /n!

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, 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 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.