Transcript Fermat’s Last Theorem: Journey to the Center of

Fermat’s Last Theorem:
Journey to the Center of
Mathematical Genius
(and Excess)
Rod Sanjabi
So, who is this Firm-at guy?
Pierre de Fermat (Fer’-mah, though some pronounce
the ‘t’), a 17th century French mathematician, is
thought of today primarily as a number theorist.
Ironically, this well-known mathematician was in fact
only an amateur in life; he was a lawyer by trade.
So, if he’s just an
amateur, what’s the big
deal?
Fermat refused to publish his work, and because of this, 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 and commentaries with the object of
publishing a notebook of his ideas. In this way, the famous
(or infamous) ’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.
OK, so what’s this
brilliant theorem?
It’s actually pretty simple:
xn+ynzn
If n>2
Or, in English, the sum of x and y raised to a certain power n
(where n is greater than 2) can equal no integer z raised to the
same power (as long as x, y, and z are all nonzero integers).
Ha! Prove it...
Now that, dear heading, is another thing altogether.
Fermat’s last theorem (stated before) proved impossible to
prove until very recently. Fermat himself said that“I have
discovered a truly remarkable proof which this margin is
too small to contain”, however, the proof has been long in
coming, and after centuries of failed attempts, has only
been provided by the use of modern mathematical methods.
Does that mean that
Fermat was lying?
Maybe. Some of the older attempted proofs include an
attempt by Euler, who wrote in a letter on 4 August 1753
claiming he had a proof of Fermat's Theorem when n = 3.
The proof he provided 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. His mistake,
nonetheless, is an interesting one, one which was to have a
bearing on later developments. He needed to find cubes of
the form p2 + 3q2. He imaginatively attempted to display
the proof using numbers of the form a+b(root 3), but he
overlooked the fact that such numbers do not behave as
integers would.
Ok...
It doesn’t make all that much sense in plain English,
but there were many other notable attempts at proving
Fermat’s Last Theorem by celebrated mathematicians
including Sophie Germain, Lejeune Dirichlet, Gabriel
Lamé (whose proof involved ‘factorizing’ the equation
into linear factors over the complex numbers), and
others.
Most notably, they all failed.
And then...
With the advent of more and more advanced mathematical
principles, proofs for Fermat’s last theorem became more
and more complex. Bernoulli’s numbers (defined by x/(ex
- 1) = Bn xn /n!) were employed to prove the theorem for
all regular prime numbers, but when it was shown that the
number of irregular (as relating to Bernoulli’s equation)
primes is infinite, the nearly-solved proof became
infinitely further from solution (pun intended).
So, finally,
The final chapter in the story began in 1955, although at the 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 ShimuraTaniyama-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.
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, at the end of which he provided the proof. In view of speculation, Wiles realized
that the proof was not completed.
Grrr...
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.”
In the end...
The proof took over 300 years to prove, and elliptic and
fractal geometry were used in the final solution...neither of
which was present in Fermat’s day. So, did he really have a
proof?
xn+ynzn...
or something.