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PARTISAN REVIEW
link in the long chain of his tightly reasoned argument by another, simpler
technique. On October 25, 1994-a date which will live in fame, at least
among mathematicians-Wiles submitted his 130-page-long, now flawless
proof of Fermat's Last Theorem for publication, and it came out in the
May 1995 issue of the
Annals of Mathematics.
The tale of Fermat's Last Theorem begins in sixth-century BCE
Greece with Pythagoras and the school of mathematicians he founded.
The Pythagoreans had shown that the equation
a
2
+
b
2
=
c
2
describes the
relation between lengths
a
and
b
of the short sides and c of the hypotenuse
of a right triangle. Whole numbers that satisfy this equation became
known as "Pythagorean triples," of which the first four are (3,4,5), (5, 12,
13), (8, 15, 17), and (7,24,25), and, as Euclid proved three centuries later,
there are an infini te number.
Diophantus of Alexandria, a Greek mathematician of the third century
CE, wrote a treatise entitled
Arithmetica
that presented many problems of
number theory, including that of the Pythagorean triples. A Latin transla–
tion,
Arithmeticorum,
was published in Paris in 1621, a copy of which was
not only bought but actually read closely by Pierre Fermat, a seventeenth–
century French upper-level civil servant (by profession) and mathematician
(by avocation).
Despite his amateur status, Fermat made many fundamental contribu–
tions to mathematics, including the development of probability theory (in
collaboration with Blaise Pascal). Rather than publishing his discoveries,
however, Fermat communicated them in private letters to friends, mostly
without including any proofs. Many of Fermat's missing proofs were later
provided by other mathematicians, foremost among them the giant of
eighteenth-century mathematics, Leonhard Euler. Fermat also made margin–
al notations in his copy of
Arithmeticorum,
jotting down some important
theorems and (sometimes) their proofS. On the page of his copy of volume
one of the six-volume treatise, in which Diophantus considers the problem
of finding Pythagorean triples (a, b, c) that satisfy the equation
a
2
+
b
2
=
c
2,
Fermat had scribbled (in Latin) what came to be known as Fermat's Last
Theorem. Its English translation, which, as any number theorist is bound
to notice, consists of twice as many words as Fermat's Latin original, reads:
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 the sum of two like powers.
