Fermat’s Last Theorem by Kelly Oakes

Pierre de Fermat (1601 – 1665) Born in Beaumont-de-Lomagne, 36 miles north- west 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: This is known as Fermat’s Last Theorem, and in more mathematical terms is: “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.” “The equation x n + y n = z n 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. From this point onwards, mathematicians tried to find proofs for classes of prime numbers, rather than just demonstrating it one number at a time... 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 x m, y m and z m 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.

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. Proofs for Special Cases 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 x n + y n = y n. 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 x 2p + y 2p = z 2p, then 2p must divide x or y.

Andrew Wiles 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.” 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. 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. 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 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 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 y 2 = (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. Elliptic curves, which have been studied since the time of Diophantus, concern cubic equations of the form:

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... a n + b n = c n is a counterexample to Fermat's Last Theorem, then the elliptic curve... y 2 = x(x - a n )(x + b n ) 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.