Transcript Fermat’s Last Theorem

Fermat’s Last
Theorem
by Kelly Oakes
Pierre de Fermat
(1601 – 1665)
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Born in Beaumont-de-Lomagne, 36 miles northwest of Toulouse, in France.
He was a lawyer at the Parlement of Toulouse,
and also an amateur mathematician (now widely
regarded as "The King of Amateurs“).
He is given credit for early developments of
calculus, his work on the theory of numbers,
inventing the proof technique of infinite descent
and Fermat’s factorisation method amongst many
other things.
During his lifetime, Fermat was known to be very
secretive and was a recluse. Very few records of
his proofs exist. In fact, many mathematicians
doubt his claims because of the difficulty of some
of the problems and the limited mathematical
tools available to him.
Fermat died in 1665, in a town called Castres, 49
miles east of Toulouse.
History
Around the year 1640, Fermat wrote, in the margin of his copy of Arithmetica
by Diophantus of Alexandria, the following:
“It is impossible to separate a cube into two cubes, or a fourth
power into two fourth powers, or in general, any power higher than
the second into two like powers. I have discovered a truly
marvellous proof of this, which this margin is too narrow to
contain.”
This is known as Fermat’s Last Theorem, and in more mathematical terms
is:
“The equation xn + yn = zn has no solution for non-zero integers
x, y, and z if n is an integer greater than 2.“
It is known as Fermat’s last theorem, not because it was Fermat’s last piece of work, but
because it was the last remaining statement that had yet to be proved or independently
verified after his death.
Proofs for Special Cases
The first case of Fermat’s Last Theorem that was solved was n=4. Fermat himself proved
that “the area of a right triangle cannot be equal to a square number”, using a method of
infinite descent. This was the only proof in number theory that Fermat left, and implies
that FLT is true when n=4.
The second case proved
(approximately one hundred years
later) was n=3, by Euler. He also used
a method of infinite descent, but his
proof was otherwise very different,
which meant that there would be no
way to generalise these two special
proofs into one that would apply to all
cases where n>2.
Leonhard Euler
Proofs for Special Cases
The proof for n=5 was created by Dirichlet and Legendre in 1825, using a
generalisation of Euler's proof for n = 3. A proof for the next prime number, n=7,
was found 14 years later by Lamé.
Generally, if we had proven FLT to any power n, then the theorem
is valid to all the multiples of n. The reason for this is that if the
numbers x, y and z are a solution for the power mn, then the
numbers xm, ym and zm are a solution to the power of n, which
contradicts the fact that FLT has been proved for the power n.
From this it follows that the only numbers left to prove FLT for are
prime, because all other numbers greater than 2 are multiples of
prime numbers.
From this point onwards, mathematicians tried to find proofs for classes of prime
numbers, rather than just demonstrating it one number at a time...
Proofs for Special Cases
In 1847, Kummer proved that the theorem
was true for all regular primes, which
includes all prime numbers below 100,
except 2, 37, 59 and 67.
Over 100 years later in 1977, Guy
Terjanian proved that if p is an odd prime
number, and the natural numbers x, y and
z satisfy x2p + y2p = z2p, then 2p must
divide x or y.
In 1983, Gerd Faltings proved the Mordell
conjecture, which implies that for any value
of n>2, there are at most finitely many
coprime integers x, y and z with xn + yn =
yn.
Andrew Wiles
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Wiles first discovered Fermat’s Last Theorem aged 10, by reading ‘The Last
Problem’ by E. T. Bell in his local library.
“It looked so simple, and yet all the great mathematicians in history couldn't
solve it. Here was a problem that I, a ten year old, could understand and I knew
from that moment that I would never let it go. I had to solve it.”
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He spent his schooldays trying to solve the problem, only stopping while he was
at Cambridge studying for his PhD on elliptic curves, a topic which later helped
him solve Fermat’s Last Theorem.
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In 1986 Wiles heard that Ken Ribet had proved that there was a link between
the Taniyama-Shimura conjecture and FLT, which meant that his childhood
ambition was now a professionally acceptable problem to work on.
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He decided that he would have to work on the problem in complete isolation, as
the interest it would create could interfere with his work, and he wanted to give it
his undivided attention.
Andrew Wiles
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Wiles after completing the proof at the
final lecture:
“ I think I’ll stop here”
In 1993, after 7 years of working on the
proof, Wiles finally believed he had
completed it. He gave a series of lectures
on the subject at the Isaac Newton
Institute in Cambridge ending on 23 June
1993.
His results were then written up for
publication, however during this process
a subtle error in a crucial part of the
argument was discovered. It seemed that
he did not have a complete proof after all.
It took over a year of work and a little help
from Cambridge mathematician Richard
Taylor, but eventually Wiles managed to
repair the error and finally complete the
proof.
The Mathematics of
Fermat’s Last
Theorem...
Elliptic Curves
Elliptic curves, which have been studied since the time of Diophantus,
concern cubic equations of the form:
y2 = (x + a).(x + b).(x + c),
where a, b & c can be any whole number, except zero.
The challenge is to identify and quantify the whole solutions to the
equations, the solutions differing according to the values of a, b,
and c.
Modular Forms
The mathematics of modular forms is much more modern than
that of elliptic curves.
Modular forms are functions that satisfy rather spectacular and
special properties resulting from their surprising array of internal
symmetries. They involve complex numbers, which are composed
of real and imaginary parts.
Taniyama-Shimura conjecture
The Taniyama-Shimura conjecture (or modularity theorem) states that every
rational elliptic curve is modular.
If...
an + b n = c n
is a counterexample to Fermat's Last Theorem, then the elliptic
curve...
y2 = x(x - an)(x + bn)
cannot be modular, thus violating the Shimura-Taniyama
conjecture.
It was Gerhard Frey that suggested that the conjecture implies Fermat's Last
Theorem, and Ken Ribet that later proved it.
In 1995, Andrew Wiles proved that the Taniyama-Shimura conjecture was true
for semistable elliptic curves, which was enough to prove Fermat’s Last
Theorem.
Further Information
Mathematics of Fermat’s Last Theorem in more depth.
Interview with Andrew Wiles.
The “Fermat Corner” on Simon Singh’s website.
The Devil and Simon Flagg by Arthur Porges – A Short Story
centred around Fermat’s Last Theorem.