164 7. ANCIENT NUMBER THEORYalready see why we need both ñ and q to be products of 2, 3, and 5. This problem
amounts to the quadratic equation .r^2 — dx — 1 =0, and its unique positive solution
is χ = d/2 + y/l + (d/2)^2. Column 1 of the tablet, which contains (d/2)^2 then
appears as part of the solution process. It is necessary to take its square root
and also the square root \Ë + d^2 /4 in order to find the solution χ = p/q. This
explanation seems to fit very well with the tablet. One could assume that the first
column gives values of d that a teacher could use to set such a problem with the
assurance that the pupil would get terminating sexagesimal expansions for both χ
and 1/x. On the other hand, it does not fully explain why the tablet gives the
numbers p^2 — q^2 and p^2 + q^2 , rather than simply ñ and q, in subsequent columns.
Doing our best for this theory, we note that columns 2 and 3 contain respectively
the numerators of χ — 1/x and ,r + l/x, and that their common denominator is
the square root of the difference of the squares of these two numerators. Against
that explanation is the fact that the Mesopotamians did not work with common
fractions. The concepts of numerator and denominator to them would have been
the concepts of dividend and divisor, and the final sexagesimal quotient would not
display these numbers. The recipe for getting from columns 2 and 3 to column
1 would be first to square each of these columns, then find the reciprocal of the
difference of the squares as a sexagesimal expansion, and finally, multiply the last
result by the square in column 2.
In the course of a plea that historians look at Mesopotamian mathematics in
its own terms rather than simply in relation to what came after, Robson (2001)
examined several theories about the purpose of the tablet and gave some imagina-
tive scenarios as to what may be in the lost portion of the tablet. Her conclusion,
the only one justified by the present state of knowledge is that "the Mystery of the
Cuneiform Tablet has not yet been fully solved."^2
And we have not claimed to solve it here. Plimpton 322 is a fascinating object
of contemplation and serves as a possible example of an early interest in what we
now call quadratic Diophantine equations. Without assuming that there is some
continuous history between Plimpton 322 and modern number theory, we can still
take quadratic Diophantine equations as a convenient starting point for discussing
the history of number theory,2. Ancient Greek number theory
Our knowledge of Pythagorean number theory is based on several sources, of which
two important ones are Books 7-9 of Euclid's Elements and a treatise on arithmetic
by the neo-Pythagorean Nicomachus of Gerasa, who lived about 100 CE. Just as
the Sun Zi Suan Jing preserves more of ancient Chinese arithmetic than the earlier
Jiu Zhang Suamhu, it happens that the treatise of Nicomachus preserves more
of Pythagorean lore than the earlier work of Euclid. For that reason, we discuss
Nicomachus first.
The Pythagoreans knew how to find the greatest common divisor of two num-
bers. A very efficient procedure for doing so is described in Chapter 13 of Book 1
of Nicomachus' Arithmetica and in Proposition 2 of Book 7 of Euclid's Elements.
This procedure, now known as the Euclidean algorithm, is what the Chinese called
the mutual-subtraction procedure. Nicomachus applies it only to integers, any two
(^2) In a posting at a mathematics history website, Robson noted that reciprocal pairs and cut-and-
paste geometry seem to be the most plausible motives for the tablet.