BOOKS
511
all that wrong in supposing that he, a twentieth-century schoolboy, knew as
much mathematics as did a seventeenth-century genius like Fermat.
After a year of getting nowhere, Wiles wisely decided that he might
be able to learn from the failures of his eminent predecessors. By the
time he was an undergraduate at Oxford, he was sophisticated enough to
realize that he was facing the same impenetrable brick wall that had faced
all the others. Yet he did not want to believe that, because it might be one
of Godel's undecidable propositions, there may not exist any proof of
FenTlat's Last Theorem. So he continued to search for a proof, supposing
that the mathematical methods for working it out were already available
and that he might be able to provide the only missing ingredient, namely
ingenuity.
By the time Wiles became a graduate student in mathematics at
Cambridge in 1979, everything he had done for the past decade had been
directed toward preparing himself for the assault on Fermat's Last
Theorem. But he realized that, being on the threshold of professional
mathematicianhood, he had to pick a research topic that would provide
him with some results by the end of three years to write up for a Ph.D.
thesis. So he put Fermat aside for a while and, on his thesis supervisor's
suggestion, set out on a study of elliptic curves (or "elliptic equations," as
Singh prefers to refer
to
them). This move proved
to
be a turning point in
Wiles's quest for his holy grail, because it put him in touch with tech–
niques that would provide an entirely novel approach to the Last Theorem.
Elliptic equations bear some resemblance to Pythagorean triplet equations,
especially as regards the challenge of finding whole number solutions, for
which reason they had been of interest already to Diophantus and Fermat.
Wiles's supervisor had suggested elliptic equations as a thesis research
topic to his Fermatomane student, not because he thought they would be
of help to Wiles in dealing with his Last Theorem obsession, but because
they had recently become very trendy. The sudden, latter-day interest in
elliptic equations arose from the work that two young Japanese mathe–
maticians at Tokyo University, Yutaka Taniyama and Goro Shimura, had
taken up in the mid-1950s. They were studying modular forms, an eso–
teric, offbeat branch of geometry developed in the nineteenth century for
the quantitative classification and description of diverse symmetric aspects
that regular objects can possess in multidimensional space. Modular forms
are expressed as algebraic series, in which each term represents an ingre–
dient of the object's overall symmetry. One day, Taniyama noticed to his
astonishment that terms of the series expressing a particular modular form
were identical to the terms of a well-known elliptic equation. Suspecting
that there might be some fundamental, albeit mysterious, connection
between these two types of algebraic series, Taniyama examined a few
