- THE MUSLIMS 179
more than one unknown. But Bhaskara also asks harder questions. For example
(Colebrooke, 1817, p. 202): Find two (rational) numbers such that the sum of
the cubes is a square and the sum of the squares is a cube. Bhaskara manages
to find a solution using the trick of assigning the ratio a of the two numbers..
It is necessary for the technique that this ratio satisfy 1 4- a^3 = b^2. Bhaskara
chooses á = 2, b = 3.^13 The smaller number is then chosen to be of the form
(1 + a^2 )w^3 for some number w. The sum of the squares will then be (1 + á^2 )^2 éõ^6 +
á^2 (1 + o?)^2 w^6 - (1 + a^2 )^3 w^6 = ((1 + a^2 )w^2 )^3 , and the sum of their cubes will be
(1 + a^2 )^3 w^9 + a^3 (l + a^2 )^3 w^9 = (1 + a^2 )^3 b^2 w^9 = 6^2 ((1 + a^2 )w^3 )^3. Hence, if w is
chosen so that (1 + a^2 )w is a square, this will be a perfect square. The simplest
choice obviously is w = 1 + a^2. In Bhaskara's example, that choice gives the pair
625 and 1250.
5. The Muslims
The Muslims continued the work of Diophantus in number theory. Abu-Kamil
(ca. 850-ca. 930) wrote a book on "indeterminate problems" in which he studied
quadratic Diophantine equations and systems of such equations in two variables.
The first 38 problems that he studied are arranged in a very strict ordering of
coefficients, exponents, and signs, making it a very systematic exposition of these
equations. Later scholars noted the astonishing fact that the first 25 of these
equations are what are now known as algebraic curves of genus 0, while the last
13 are of genus 1, even though the concept of genus of an algebraic curve is a
nineteenth-century invention (Baigozhina, 1995).
Muslim mathematicians also went beyond what is in Euclid and Nicomachus,
generalizing perfect numbers. In a number of articles, Rashed (see, for example,
1989) points out that a large amount of theory of abundant, deficient, and perfect
numbers was assembled in the ninth century by Thabit ibn-Qurra and others, and
that ibn al-Haytham (965-1040) was the first to state and attempt to prove that
Euclid's formula gives all the even perfect numbers. Thabit ibn Qurra made an in-
teresting contribution to the theory of amicable numbers. A pair of numbers is said
to be amicable if each is the sum of the proper divisors of the other. The smallest
such pair of numbers is 220 and 284. Although these numbers are not discussed
by Euclid or Nicomachus, the commentator Iamblichus (see Dickson, 1919, p. 38)
ascribed this notion to Pythagoras, who is reported as saying, "A friend is another
self." This definition of a friend is given by Aristotle in his Nicomachean Ethics
(Bekker 1170b, line 7).
We mentioned above the standard way of generating perfect numbers, namely
the Euclidean formula 2n_1(2n - 1), whenever 2" - 1 is a prime. Thabit ibn-Qurra
found a similar way of generating pairs of amicable numbers. His formula is
2"(3 · 2n - 1)(3 · 2n_1 - 1) and 2n(9 · 22n_1 - 1),
whenever 3 • 2 n - 1, 3 · 2n_1 - 1, and 9 • 2^2 "-1 - 1 are all prime. The case ç = 2
gives the pair 220 and 284. Whatever one may think about the impracticality
of amicable numbers, there is no denying that Thabit's discovery indicates very(^13) It was conjectured in 1844 by the Belgian mathematician Eugene Charles Catalan (1814-1894)
that the only nonzero solutions to the Diophantine equation pm — qn = 1 are g = 2 = ç, ñ = 3 — n.
This conjecture was proved in 2002 by Predhu Mihailescu, a young mathematician at the Institute
for Scientific Computing in Zurich. Lucky Bhaskara! He found the only possible solution.