170 7. ANCIENT NUMBER THEORYof constructing perfect numbers (Proposition 36), quoted above. No perfect num-
ber has yet been found that is not generated by this procedure, although no proof
exists that all perfect numbers are of this form. Any exception would have to be
an odd number, since it is known (see Problem 7.8) that all even perfect numbers
are of this form.
From the modern point of view, Euclid's number theory is missing an explicit
statement of the fundamental theorem of arithmetic. This theorem, which asserts
that every positive integer can be written in only one way as a product of prime
numbers, can easily be deduced from Book 7, Proposition 24: // two numbers are
relatively prime to a third, their product is also relatively prime to it. However,
modern historians (Knorr, 1976) have pointed out that Euclid doesn't actually
prove the fundamental theorem.
2.3. The Arithmetica of Diophantus. Two works of Diophantus have survived
in part, a treatise on polygonal numbers and the work for which he is best known,
the Arithmetica. Like many other ancient works, these two works of Diophantus
survived because of the efforts of a ninth-century Byzantine mathematician named
Leon, who organized a major effort to copy and preserve these works. There is
little record of the influence the works of Diophantus may have exerted before this
time.
According to the introduction to the Arithmetica, this work consisted originally
of 13 books, but until recently only six were known to have survived; it was assumed
that these were the first six books, on which Hypatia wrote a commentary. However,
more books were recently found in an Arabic manuscript that the experts say is
a translation made very early probably in the ninth century. Sesiano (1982)
stated that these books are in fact the books numbered 4 to 7, and that the books
previously numbered 4 to 6 must come after them.
Diophantus begins with a small number of determinate problems that illustrate
how to think algebraically, in terms of expressions involving a variable. Since these
problems belong properly to algebra, they are discussed in Chapter 14. Indeter-
minate problems, which are number theory because the solutions are required to
be rational numbers (the only kind recognized by Diophantus), begin in Book 2.
A famous example of this type is Problem 8 of Book 2, to separate a given square
number into two squares. Diophantus illustrates this problem using the number 16
as an example. His method of solving this problem is to express the two numbers
in terms of a single unknown, which we shall denote ò, in such a way that one of
the conditions is satisfied automatically. Thus, letting one of the two squares be
ò^2 , which Diophantus wrote as Äõ (as explained in Chapter 14), he noted that the
other will automatically be 16 -ò^2. To get a determinate equation for ò, he assumes
that the other number to be squared is 4 less than an unspecified multiple of ς. The
number 4 is chosen because it is the square root of 16. In our terms, it leads to a
quadratic equation one of whose roots is zero, so that the other root can be found by
solving a linear equation. As we would write it, assuming that 16 — ò^2 = (fee — 4)^2 ,
wc find that (k^2 + 1)ς^2 = 8kς, and—cancelling ς, since Diophantus does not operate
with 0 we get ς = 8k/(k^2 + 1). This formula generates a whole infinite family of
solutions of the equation that we would call x^2 + y^2 = 16 via the identity