Code crackers
Inside the minds of the maths maestros
When Andrew Wiles received the £500,000 Abel Prize for mathematics last week, there was a general sense of “At last!” in the mathematical community. After all, Professor Wiles had already won almost every other prize for his 1995 proof of Fermat’s last theorem, the most notorious problem in the history of mathematics.
As it has been mentioned in Doctor Who, Star Trek: the Next Generation,
The Simpsons and the Liz Hurley blockbuster Bedazzled, I would hope that most people would know the intricacies of Fermat’s last theorem by now, but here’s a quick recap for those who are still puzzled about why there is so much fuss over solving a maths problem.
The story starts with Pierre de Fermat, a French judge and one of the all-time great mathematicians, who in 1637 claimed he could prove that the equation (an + bn = cn) has no whole number solutions when n is greater than 2. There are some near misses (eg, 63 + 83 = 93 – 1), but no numbers that make the equation balance properly.
Given that there are many possible numbers to check, it was quite a claim, but Fermat was absolutely sure that no numbers fitted the equation because he had a logical watertight argument. Sadly, he never wrote down his proof. Instead, in the margin of a book, he left a tantalising note in Latin: “I have a truly marvellous demonstration of this proposition ( demonstration-em-mirabilem), which this margin is too narrow to contain.”
After Fermat’s death, mathematicians found lots of similar notes (“I can prove this, but I have to feed the cat” or “I can prove that, but I have to wash my hair”), so they set about rediscovering Fermat’s supposed proofs. They were successful in every case, with the exception of proving that (an + bn = cn) has no solutions, which is why it became known as Fermat’s last theorem, namely the last one that could be proven.
For the next three centuries, mathematicians tried and failed to find a proof, which is why Wiles’s eventual success was such a major achievement, and why he has been showered with prizes and accolades. For example, there was the King Faisal International Prize (£140,000), the Wolf Prize (£70,000), a knighthood, and the Oxford maths department is now housed in the Andrew Wiles Building. It was even rumoured that Gap asked him to endorse its range of menswear.
Wiles worked in secrecy for seven years before revealing his 200-page proof
The most valuable prize should have been the Wolfskehl prize: 100,000 German marks bequeathed by Paul Wolfskehl in 1906. It is said that the wealthy German industrialist was about to take his own life after a failed romance, but an encounter with Fermat’s last theorem made him fall in love with mathematics and his new-found passion gave him the will to live. When Wiles won the Wolfskehl prize in 1997, it should have been worth at least £1 million, but massive hyperinflation in Germany after the First World War meant that Wiles received a relatively paltry £30,000.
Of course, the money is irrelevant, because Wiles was driven by the desire to answer one of history’s greatest riddles. He had stumbled upon the last theorem as a 10-yearold and then spent the next 30 years working on the problem. A childhood dream evolved into an adult obsession, and when he eventually figured out a possible strategy for proving Fermat’s riddle, he worked in secrecy for seven years before revealing his 200-page proof.
Every mathematician that I have ever met takes on these major problems purely for the intellectual battle, and the rich prizes are just a distraction, usually accepted, but sometimes rejected.
In 2000, the Clay Mathematics Institute identified seven great mathematical mysteries and offered a $1 million reward for each of these so-called Millennium Problems. So far, only the Poincaré conjecture has fallen. Posed in 1904 by Henri Poincaré, the conjecture suggests that “every simply connected, closed 3-manifold is homeomorphic to the 3-sphere”, and Grigori Perelman from Saint Petersburg successfully proved that it is indeed true. When he was offered the £1 million prize, he declined it.
Perelman was then offered the Fields Medal, the most prestigious prize in mathematics. It is only open to those 40 years old or younger, so it is just about the only prize that has eluded Andrew Wiles. Perelman was exactly 40, but he decide to spurn the prize, even after Sir John Ball, President of the International Mathematical Union, flew to St Petersburg and spent 10 hours trying to persuade him to accept the award.
Perelman later recounted what happened during that visit: “He proposed to me three alternatives: accept and come; accept and don’t come, and we will send you the medal later; third, don’t accept the prize. From the very beginning, I told him that I have chosen the third one... [the prize] was completely irrelevant for me. Everybody understood that if the proof is correct, then no other recognition is needed.”
Andrew Wiles graciously accepts the medals and the cheques, but agrees with Perelman that there are other much richer prizes: “I had this very rare privilege of being able to pursue in my adult life what had been my childhood dream. I know it’s a rare privilege, but if you can tackle something in adult life that means that much to you, then it’s more rewarding than anything imaginable.” Simon Singh is the author of Fermat’s
Last Theorem, the first book about mathematics to become a number one bestseller