x2+y2=z2. This is the Pythagoras’ theorem, a simple theorem that a child would recognize. However, just a slight change to the above equation would transform it to the world’s hardest mathematical problem that confounded the world’s greatest mathematicians for the last 358 years; this is the problem of proving Fermat’s Last Theorem.
Fermat’s Last Theorem proposed that one would never find any solution if the degree ‘2’ in the above equation is replaced by any whole number greater than 2. For example, there would be no solution for the equation x3+y3=z3. Fermat scribbled in the margin of a book saying that, ‘I have discovered a truly marvellous proof, which this margin is too small to contain.’
This small note in the margin had given an ultimate challenge to mathematicians for the next 300 years. This particular problem is very special as Fermat said that he had a proof, but it was never revealed. Many of the world’s greatest minds had spent a lot of time on this problem, but they did not succeed in finding a proof.
Finally, this theorem was proved in 1995 by Andrew Wiles, a professor at the Princeton University. After seven years of complete isolation and secrecy, he had fulfilled his childhood dream at last. How did Professor Wiles actually solve the world’s most difficult mathematical problem?
Professor Andrew Wiles
Elliptic curves, modular forms
and the Taniyama-Shimura conjecture have a profound importance in this proof.Elliptic
curves are cubic curves whose solution have a shape that looks like a torus (a doughnut);
the equation y2=x3+ax2+bx+c represents an elliptic curve.
Modular forms are functions on the complex plane that are inordinately symmetric. The
Taniyama-Shimura conjecture suggests that every elliptic curve is a modular form in
disguise.
An elliptic curve
Gerhard Frey, a German mathematician, started wondering what would happen if Fermat was wrong and there was a solution to the equation. With the work of another mathematician Ken Ribet, it was proved that if there were a solution to the Fermat equation, an elliptic curve which was not modular would result. This violated the Taniyama-Shimura conjecture.
An example of a modular form
On the contrary, if the Taniyama-Shimura conjecture is correct, then Fermat’s Last Theorem could be proved as the above non-modular elliptic curve implies that a solution to Fermat’s equation could not exist.
Thus, to prove Fermat’s Last Theorem is to prove that the Taniyama-Shimura conjecture is correct, i.e. every elliptic curve is modular. This was when Andrew Wiles started his work. He abandoned all his other research and locked himself up for 7 years, determined to find a proof to Fermat’s Last Theorem.
One method to prove that every elliptic curve is modular is to prove that there is the same number of elliptic curves and modular elliptic curves. In order to compare and count more easily, Wiles transformed the elliptic curves into Galois representations and try to compare Galois representations with modular forms. He tried to use the Iwasawa theory to help him with the counting mechanism and create a class number formula, but he failed. Finally, with the help of Flach’s class number formula, Wiles thought that he had succeeded in proving that every elliptic curve was modular. Since the Taniyama-Shimura conjecture seemed to be proved, he thought he had proved Fermat’s Last Theorem.
After seven years of hard work,
Wiles presented his proof of Fermat’s Last Theorem to the world in Cambridge in June
1993 under a suspenseful atmosphere. After his lecture, there were enquiries from
newspapers and journalists from all around the world immediately. The only thing left was
to send the proof to be refereed (to be read by other mathematicians to see if the proof
was really correct or not.) Unexpectedly, there was a fundamental error.
Professor Wiles explaining his proof
Wiles paid every effort in the next year trying to fix the problem arising from the method of Flach. It was extremely difficult as it was like putting a large carpet in a small room. If he put the carpet smoothly in one corner, the carpet would pop up in another corner. After a year of unsuccessful attempt, Wiles was ready to abandon his flawed proof and before that, he decided to re-visit his proof for one more time. In that particular September morning when Wiles reviewed his flawed proof, an incredible revelation came to him and he realized what was blocking his method of Flach was what would make Iwasawa theory work, the theory that he had abandoned three years ago. Wiles stared at his simple and elegant proof in disbelief for twenty minutes. He finally completed the proof of the century – the proof of Fermat’s Last Theorem, a mathematical problem which had confounded the world’s greatest minds for 358 years.