% History of Fermat's Last Theorem
% by Andrew Granville
% A version edited by Ian Katz appeared in The Guardian
% (U.K.), on June 24
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It was announced yesterday that the most famous
question in pure mathematics, Fermat's Last Theorem,
has finally been answered. Englishman Andrew Wiles,
of Princeton University, a Fellow of the Royal Society,
told an academic audience at the
Isaac Newton Institute in Cambridge, including
leading authorities from around the world,
that he had finally concluded one of the most
controversial chapters in scientific history.
The story of Fermat's Last Theorem can be traced back
to ancient Greek times and stems from one of the best-known results in
mathematics, Pythagoras' Theorem. That is, in any triangle
which contains a right-angle, the square of the length of the
hypoteneuse is equal to the sum of the squares of the lengths
of the other two sides. (In mathematical notation,
$z^2 = x^2 + y^2$, where $z$ is the length of the hypotenuse
and $x$ and $y$ are the lengths of the other two sides.)
This rule was extremely useful to
the ancient Greeks because it allowed them
to construct an accurate right-angle. The idea was simply to
cut pieces of rope of lengths $x, y$ and $z$ feet, pull the ropes
taut (holding the ends together) and, voil\`a, a right angle.
For this to be practical they needed $x, y$ and $z$ to each be whole
numbers; and the example $5^2=3^2+4^2$ sufficed for their purposes.
Other examples that they recorded included $13^2=5^2+12^2$ and
even $8161^2=4961^2+6480^2$.
One of the great intellectual masterpieces of the ancient Greek
world was Diophantus' {\sl Arithmetic}. This work, available in Latin
translation in the seveteenth century, was an important
inspiration for the scientific renaissance of that period, read by
Fermat, Descartes, Newton and others.
Fermat, a jurist from Toulouse, studied mathematics as a hobby. He
didn't formally publish his work but rather disseminated his ideas
in letters, challenging others to match and/or admire his understanding.
Fermat was evidently inspired by the {\sl Arithmetic} and made many
notes in the margin of his copy. After Fermat's death, his son
Samuel published these notes and amongst them was the following
tantalizing sentence, beside the description of
Pythagoras' Theorem: ``...it is impossible for a cube to be
written as a sum of two cubes or a fourth power to be written
as a sum of two fourth powers or, in general, for any number
which is a power greater than the second to be written as
a sum of two like powers.
I have a truly marvelous demonstration of this proposition which this
margin is too narrow to contain''.
In mathematical notation, one cannot
find whole numbers $x,y,z$ and $n$, with $n$ bigger than $2$, for which
$z^n = x^n + y^n$. Whether Fermat was
being overly optimistic about his `demonstration', we
shall probably never know, but his argument has not been reproduced
in the intervening three and a half centuries, despite
no shortage of effort to do so.
His remarkably simple, seductively simple, assertion is known as
{\sl Fermat's Last Theorem}. Substantial prizes have been offered
for its solution (the largest of which, the Wolfskehl prize,
was rendered trivial by the
German hyper-inflation of the twenties). The great mathematicians of history
have been in two minds as to whether to work on it. Ernst Kummer,
the German mathematician of the last century who did so much to
establish modern algebra, wrote that Fermat's Last Theorem is
``more of a joke than a pinnacle of science'', yet his
most important work originated in his failed attempts to prove it !
Fermat's Last Theorem is the question that mathematicians love to hate:
it is really only one example of an equation, and then one that isn't so
relevant in a general study of equations. On the other hand, there is no
denying its charm and simplicity. As a question it goes in and out of
favour, sometimes being dismissed as `irrelevant' and `unimportant',
at other times hailed as the most interesting equation around.
Many people have tried to give an unassailable `proof' that Fermat's Last Theorem is true. It is a favourite of professional and
amateurs alike: and, every few years, the newspapers
report yet another purported proof, which is subsequently
found to be lacking in some way.
So why is Professor Wiles's attempt
different ? And why are there many of us, who are usually extremely skeptical
of such claims, prepared to believe
that Wiles could have succeeded where so many others have failed ?
To understand, it helps to have some perspective of the recent history
of the problem: \ Up until the last decade the question seemed unassailable.
To be sure, many interesting approaches
have been proposed, persuading us of the truth of
Fermat's Last Theorem, but without providing a proof that can be checked
in every case. One such `proof' is known to work for every
exponent $n$ up to four million: that is, for every number $n$ up to four
million there are no
whole numbers $x, y$ and $z$ for which $z^n=x^n+y^n$ has solutions.
However this method of proof involves an individual computer verification
for each $n$, and so will not give the result for {\bf all} exponents $n$.
In 1983, a young German mathematician, Gerd Faltings,
proved a result far, far beyond what anyone had thought provable.
Instead of looking only at the equations $z^n=x^n+y^n$ in isolation,
he studied a very wide class of equations, the so-called `{\sl curves}', and
discovered exactly when they could and could not have infinitely many solutions
in whole numbers. In particular,
Faltings almost proved Fermat's Last Theorem; he showed
that for any given $n>2$ there are only a few, that is finitely many,
whole number solutions to $z^n=x^n+y^n$ (not allowing `scaling'):
however, it seems
unlikely that his methods can be modified to actually show that there
are {\sl no} solutions. Faltings' work revolutionized the way that
people thought about solving equations, leading to the award of a
Fields' Medal, the mathematical equivalent of the Nobel Prize.
Wiles' approach to Fermat's Last Theorem comes, however, from a somewhat
different direction, with rather different types of ideas. It all
began in 1955, with a question posed by the Japanese mathematician
Yutaka Taniyama:
\ Could one explain the properties of {\sl elliptic curves},
equations of the form $y^2=x^3+ax+b$ with $a$ and $b$ given whole numbers,
in terms of a few well-chosen curves. That is, is there some very special
class of equations that in some way encapsulate everything there is
to know about our elliptic curves ? Taniyama was fairly specific about
these very special curves (the so-called {\sl modular curves})
and in 1968, Andr\'e Weil, brother
of the philosopher Simone Weil, and himself one of the leading mathematicians
of the century, made explicit which modular curve should describe which
elliptic curve. In 1971 the first significant proven evidence in favour of
this abstract understanding of equations was given by Goro Shimura,
a Japanese mathematician at Princeton University, who
showed that it works for a very special class of equations.
This somewhat esoteric proposed approach to understanding elliptic curves,
is now known as the Shimura-Taniyama-Weil conjecture.
There the matter
stood until 1986 when another young German researcher, this
time Gerhard Frey from Saarbr\"ucken, made the most surprising and
innovative link
between this very abstract conjecture and Fermat's Last Theorem.
What he realized was that if $c^n=a^n+b^n$ then it seemed unlikely that
one could understand the equation $y^2=x(x-a^n)(x+b^n)$ in the way
proposed by Taniyama. It took deep and difficult reasoning by Jean-Pierre
Serre in Paris, and Ken Ribet in Berkeley, California, to strengthen
Frey's original concept to the point that a counterexample to
Fermat's Last Theorem would directly contradict
the Shimura-Taniyama-Weil conjecture.
This is the point where Wiles enters the picture. Wiles, already
established as one of the deepest pure mathematicians of
his generation, has drawn together a vast array of techniques
to attack this question. Motivated by extraordinary new methods of
Victor Kolyvagin of the Sieklov Institute in Moscow, and Barry Mazur of
Harvard Univeristy, Wiles has
succeeded in establishing the Shimura-Taniyama-Weil conjecture
for an important class of examples, including those relevant to
proving Fermat's Last Theorem. His work can be viewed as a
blend of arithmetic and geometry, and has its origins way back in
Diophantus' Arithmetic. However he
employs the latest ideas from a score of different fields, from
the theories of $L$-functions, group schemes, crystalline cohomology,
Galois representations, modular forms,
deformation theory, Gorenstein rings, Euler systems and many others.
He uses, in an essential way, concepts due to mathematicians from Britain,
France, Germany, Italy, Japan, the United States, Canada, Russia and
Colombia; the culmination of work by many people around the world thinking
about very different questions. Few suspected that their work might have
this kind of application; and there are perhaps no more
than half-a-dozen people in the world who are capable of fully
understanding all the details of what Wiles has done.
Given the enormous complexity of this work it will take some time
to be certain that every detail Wiles relies upon,
from such an array of areas, is correct. However, leading experts have now
examined the essential ideas and there can be no question
that his work is a profound contribution which sheds
light on many important questions in pure mathematics. It is
a tour de force, and will stand as one of the scientific achievements
of the century. His work is not to be seen in isolation, but
rather as the culmination of much recent thinking in many directions.
There can be little doubt that over the next few years, mathematicians
will shorten and simplify Wiles' proof, so that it will be accessible
to a wider audience. (Wiles' proof, starting from scratch, would surely
be over a thousand pages long).
The story of this important discovery is a tribute to the deeper and
more abtruse levels of abstract understanding that mathematicians have long
claimed as essential. Many of us, while hailing Wiles' magnificent achievement,
yearn for Fermat to have been correct, and for the truly marvellous, and presumably comparatively straightforward, proof to be recovered.
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