Andrew John Wiles (1954 - )

Andrew Wiles is one of mathematics heroes. He is known as the man who proved Fermat’s Last Theorem. He first stumbled across the theorem in a library at the tender age of ten, little knowing that it would shape his life.

Wiles worked largely in isolation, which is rare (but not completely unknown) for a mathematician. After his proof he said:

“… after a few years I realised that talking to people casually about Fermat was impossible because it generated too much interest and you cannot focus yourself for years unless you have this kind of undivided concentration which too many spectators would destroy…”

Clearly, his was an awesome undertaking, but the way that he carried it through provides a powerful insight into the nature of mathematical discovery - relevant to pupils at school as it is to researchers at the boundaries of the subject:

“Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it’s completely dark.

“You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated.

“You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of — and couldn’t exist without — the many months of stumbling around in the dark that proceed them.”

Using a number of other results Wiles was ultimately in a position to prove not the full version of the Shimura-Taniyama conjecture, a result followed through from the theorem, but a version sufficient for his purposes. In other words that the elliptic curve,

y^2=x(x-a^n)(x+b^n)

is modular, and this proves Fermat’s Last Theorem.

When Wiles completed his proof he decided to announce it during a series of lectures that he had agreed to give in Cambridge. He decided, fittingly, that the concluding lecture would be devoted to his proof. That was received with the rapture and acclaim that it deserved.

But then, when he came to write it up, a problem emerged; even worse he now had to fix the problem in the glare of the worlds’ mathematicians and the media. This was an almost insurmountable barrier at the end of a very long and otherwise lonely road.
He spent another year with a trusted colleague Richard Taylor, again in isolation, working on the proof. He finally achieved his goal on 19th September 1994, when he irrefutably proved Fermat’s Last Theorem.

He published his proof in the Annals of Mathematics in 1995 - Modular elliptic curves and Fermat’s Last Theorem - and has since been awarded numerous prizes on account of his outstanding work.

Sadly at 41, he was too old to receive the Fields Medal - winners must be younger than forty - but instead he was awarded a special silver plaque at the Fields Medal ceremony in honour of his incredible achievement. The Fields Medal is the mathematical equivalent of the Nobel prize.

Wiles’s mathematics

Fermat’s theorem is based on a generalisation of Pythagoras’ Theorem that relates the sides of a right-angled triangle,

a^2+b^2=c^2

This has an infinite number of whole number solutions: (3,4,5), (5,12,13), …

Its generalisation,

a^n+b^n=c^n

with n > 2, was asserted by Fermat to have no solutions at all in whole numbers. It remained an assertion for over 300 years - no proof was forthcoming, but neither was a counterexample found.

Now we fast forward to the 1980s. Wiles had by now studied mathematics at Cambridge, completed his doctorate in Number Theory and was at Princeton, in the company of the worlds’ finest mathematicians.

He learnt that two other mathematicians had made a breakthrough in Fermat’s Last Theorem. Gert Frey (from Germany) and Ken Ribet (from America) had proved that the theorem followed from a result called the Shimura-Taniyama conjecture. Translated into simple (ish) language, this means that 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, and this would violate the Shimura-Taniyama conjecture. This gave Wiles just the opportunity he had been waiting for - and for the next seven years, he devoted all of his professional (and much of his personal) life to seeking a proof of the The Shimura-Taniyama conjecture knowing that a proof of Fermat’s Last Theorem would automatically follow.
Although Wiles has disposed of Fermat’s Last Theorem it still leaves a proof of the full Shimura-Taniyama conjecture open - that remains (2004) ongoing. However, there are many puzzling relations that deceptively mimic Fermat. For example:

3^3+4^3+5^3=6^3 is a classic result;

30^4+120^4+272^4+315^4=353^4 was found by Norrie in 1911;

27^5+84^5+110^5+133^5=144^5 was found by Lander and Parkin in 1966;

95800^4+217519^4+414560^4=422481^4 was found by Elkies and Frye in 1988.

There are also numerous ‘near misses’ to Fermat - Noamh Elkies has found the following:

328658^6+1507253^6=(0.99999999999999999990 \ldots )1507280^6

3472073^7+4627011^7=(1.00000000000000000000036 \ldots )4710868^7

386692^7+411413^7=(0.9999999999999999989 \ldots )441849^7

21860^9+25208^9=(0.999999999999979 \ldots )25903^9

280^{10}+305^{10}=(0.999999997 \ldots )316^{10}

434437^{15}+588129^{15}=(0.999999999999999998 \ldots )588544^{15}

1803664^{16}+2298565^{16}=(1.00000000000000000011 \ldots )2301505^{16}

There are many outstanding problems in mathematics, few have the appeal and simplicity that Fermat’s Last Theorem had.