The Proof
I was invited by the Computational Anatomy group at the Medical Imaging Analysis Lab to see a PBS documentary on Fermat’s Last theorem. Fermat, a lawyer by profession, was also a prolific mathematician. He left behind one of the hardest problems in mathematics.
Mathematically, his theorem stated that for the equation:
is unsolvable for the case n>2 (a, b, c and n are integers.) For the n=2 case, the equation is simply the Pythagoras theorem. He also left behind a note which said that he had a proof for statement, but the margin was too small to contain it.
The solution to the problem was done by Dr. Andrew Wiles from Princeton using 20th century mathematics. It’s highly unlikely that Fermat had thought of the same proof.
The key to his proof were in the connections between elliptical curves and modular forms. Elliptical curves are equations of the form:
The equations are non-singular (don’t intersect.) Cryptography based on elliptical curves have been proposed to replace RSA because the problem is fully exponential, unlike prime factoring which is sub-exponential.
Back to our story…
Right after the second world war, two Japanese mathematicians Shimura and Taniyama proposed a conjecture that elliptical curves are just modular forms in disguise (known as the Shimura-Taniyama theorem.) Think of modular forms as highly symmetric functions.
Decades passed without anyone making any breakthroughs in proving the Shimura-Taniyama conjecture, but that didn’t stop people from developing math atop it. Then, in the mid-80s, Dr. Kenneth Ribet from UC Berkeley made a connection between the conjecture and Fermat’s theorem.
His argument is as follows: If somebody were to find a set of numbers that satisfies Fermat’s equation, then this set of numbers could be used to construct an elliptical curve that is not modular, thus disproving the Shimura-Taniyama conjecture.
Working backwards from the graphic, if the Shimura-Taniyama conjecture is true, then all elliptical curves are modular, thus with no solution to Fermat’s equation, making it true.
The rest of the documentary was about how Dr. Wiles took on the challenge to prove the Shimura-Taniyama conjecture. Solving Fermat’s last theorem had been his life-long ambition. He worked for many years in isolation and when he finally proposed a solution, it was found to have a mistake. Disappointed, he re-traced his steps and proposed an alternative argument to the solution. He had solved the hardest problem in math and become a part of history.
I think this documentary was more about ambitions, dreams and aspirations of a person than the mathematics. Dr. Wiles took upon this monumental task of achieving his childhood dream. Fame, money and prestige were unimportant to him.
In our discussion that ensued, it was clear that major breakthroughs can no longer happen by one person alone. One of the scientists in the documentary read out a list of at least 30 people who directly contributed to the proof.
November 20th, 2006 at 3:52 pm
Have you looked at Fermat’s Little Theorem? It’s an interesting way of looking at divisibility criteria and testing for primality.
You should also read (if you have not already) Simon Singh’s book on this topic (in the US, it’s called Fermat’s Last Enigma and it’s called FLT elsewhere).
Loads of fun!
November 20th, 2006 at 11:17 pm
Thanks for pointing out Fermat’s Little Theorem - haven’t come across it before.
I have read one of Simon Singh’s other books - The Code Book. I used it a lot for a paper I had to write on the social aspects of cryptology and privacy.