1997*Fermat's Last Theorem*.*The Code Book*. 1999

It is
impossible for a cube to be written as a sum of two cubes or a fourth
power to be written as the sum of two fourth powers, or, in general,
for any number which is a power greater than the second to be written
as a sum of two like powers. I have a truly marvellous demonstration
of this proposition which this margin is too narrow to contain.

Fermat wrote those words 350 years ago, and ever since his famous unproved "Last Theorem" has been a cause of strong fascination, and even stronger frustration, as mathematicians, professional and amateur alike, have failed to prove it. In more modern terminology, Fermat's statement is:

x+^{n}y=^{n}zhas no whole number solutions for^{n}n> 2

The BBC's *Horizon* is a TV series about modern scientific and
technical advances. A recent programme was dedicated to Andrew Wiles'
historic proof of Fermat's Last Theorem, and, perhaps surprisingly, it
made fascinating viewing. Singh, one of the team who made the film,
here writes up the story in more detail. He weaves together two
stories: that of the theorem and the mathematical developments it has
catalysed, and that of Wiles' developing passion for it. There are
also the usual illustrations of the relevant mathematicians, along
with facsimiles of the Annotated *Arithmetica*, with Fermat's
famous marginalia, of Galois' frantic pre-duel scribblings, and of the
first page of Wiles' proof (totally incomprehensible, of course).

Much of the early material should be familiar to anyone who has read
a few history of mathematics books, but here it is told in more
detail, as a single tale of one theorem, with a sense of mounting
excitement, because *this* time we know there is to be a "happy
ending". The latter half has the new material, and new drama. The
story couldn't be more exciting if it had been scripted as a thriller:
"One man alone on a quest, triumphantly gains his goal, and the
story is over. But no! In a final twist, that goal proves false. A
last desperate lunge, and the true goal is attained."

Some of the more peripheral material seems to me to be padding. But the main story is well told, deeply fascinating, with a genuine feeling of excitement -- I would have liked more of it, especially more technical detail. Any mathematical detail is relegated to Appendices, but even these are not as deep as many popularised mathematics books.

We get a real sense of excitement, and of elation, as the proof is
finally completed. This is tempered with just a small sense of
disappointment, that this 350-year-old quest, that has fired the
imaginations of so many, is finally over. (The proof might never have
come about had Wiles not had the opportunity, supplied at Princeton,
to spend seven isolated *years* working on nothing else -- a
lesson for today's "efficiency" ideas of requiring "industrial
management" practices to be applied to university research.)

Singh makes clear is that, in order to do his >100 page proof of
Fermat's Last Theorem, Wiles used much apparatus from modern number
theory, and also produced much new deep mathematics himself. This
proof does not just fill in a small, obscure, 350-year-old hole; it is
a tremendous advance in number theory in its own right. Mathematicians
are excited by it because it is actually a proof of the *Taniyama-Shimura
conjecture* (which others had earlier proved implies Fermat's Last
Theorem), and that conjecture is used as the basis for much other work
in number theory.

Clearly, Wiles' 20th century proof cannot be Fermat's own alleged
proof, restricted to 17th century mathematics. If Fermat really did
have a proof, it must have been much simpler, and there is still a
chance for someone to discover it. But if Fermat's own proof was
flawed, as many believe, we should be thankful his copy of Diophantus'
*Arithmetica* didn't have wider margins. For then the flaw would
have been spotted very quickly, and this wonderful puzzle might never
have resulted in such great mathematical advances.