- ANCIENT GREEK NUMBER THEORY 171
You may be asking why it was necessary to use a square number (16) here. Why
not separate any positive rational number, say 5, into a sum of two squares? If
you look carefully at the solution, you will see that Diophantus had to make the
constant term drop out of the quadratic equation, and that could only be done by
introducing the square root of the given number.
Diophantus' procedure is slightly less general than what we have just shown,
although his illustrations show that he knows the general procedure and could
generate other solutions. In his illustration he assumes that the other square is
(2ò - 4)^2. Since this number must be 16 - ò^2 , he finds that 4ò^2 - 16ò +16 = 16 - ò^2 ,
so that ò = ø. It is clear that this procedure can be applied very generally, showing
an infinite number of ways of dividing a given square into two other squares.
At first sight it appears that number theory really is not involved in this prob-
lem, that it is a matter of pure algebra. However, the topic of the problem naturally
leads to other questions that definitely do involve number theory, that is, the the-
ory of divisibility of integers. The most obvious one is the problem of finding all
possible representations of a positive rational number as the sum of the squares of
two rational numbers. One could then generalize and ask how many ways a given
rational number can be represented as the sum of the cubes or fourth powers, and
so forth, of two rational numbers. Those of a more Pythagorean bent might ask
how many ways a number can be represented as a sum of triangular, pentagonal,
or hexagonal numbers. In fact, all of these questions have been asked, starting in
the seventeenth century.
The problem just solved achieved lasting fame when Fermat, who was studying
the Arithmetica, remarked that the analogous problem for cubes and higher powers
had no solutions; that is, one cannot find positive integers x, y, and æ satisfying
x^3 + y^3 = z^3 or x^4 + y^4 = z^4 , or, in general xn +yn = zn with ç > 2. Fermat stated
that he had found a proof of this fact, but unfortunately did not have room to write
it in the margin of the book. Fermat never published any general proof of this fact,
although the special case ç = 4 is a consequence of a method of proof developed by
Fermat, known as the method of infinite descent. The problem became generally
known after 1670, when Fermat's son published an edition of Diophantus' work
along with Fermat's notes. It was a tantalizing problem because of its comprehen-
sibility. Anyone with a high-school education in mathematics can understand the
statement of the problem, and probably the majority of mathematicians dreamed
of solving it when they were young. Despite the efforts of hundreds of amateurs
and prizes offered for the solution, no correct proof was found for more than 350
years. On June 23, 1993, the British mathematician Andrew Wiles announced at
a conference held at Cambridge University that he had succeeded in proving a cer-
tain conjecture in algebraic geometry known as the Shimura-Taniyama conjecture,
from which Fermat's conjecture is known to follow. This was the first claim of a
proof by a reputable mathematician using a technique that is known to be feasible,
and the result was tentatively endorsed by other mathematicians of high reputa-
tion. After several months of checking, some doubts arose. Wiles had claimed in
his announcement that certain techniques involving what are called Euler systems
could be extended in a particular way, and this extension proved to be doubtful. In
collaboration with another British mathematician, Richard Taylor, Wiles eventu-
ally found an alternative approach that simplified the proof considerably, and there
is now no doubt among the experts in number theory that the problem has been
solved.