Number Theory in Asia

Preview

In the next three chapters we see algebra, in the form of techniques for manipulating equations, becoming firmly established in mathematics. The present chapter shows equations applied to number theory, Chapter 6 shows equations studied for their own sake, and Chapter 7 shows equations applied to geometry. As we saw in Chapter 3, Diophantus had methods for finding rational solutions of quadratic and cubic equations. But when integer solutions are sought, even linear equations are not trivial. The first general solutions of linear equations in integers were found in China and India, along with independent discoveries of the Euclidean algorithm. The Indians also rediscovered Pell’s equation $x^2 - Ny^2 = 1$, and found methods of solving it for general natural number values of N. The first advance on Pell’s equation was made by Brahmagupta, who in 628 ce found a way of “composing” solutions of $x^2 - Ny^2 = k_1$ and$x^2 - Ny^2 = k_2$ to produce a solution of $x^2 - Ny^2 = k_1k_2$. (We also touch on a curious formula of Brahmagupta that gives all triangles with rational sides and rational area.) In 1150 ce, Bhâaskara II found an extension of Brahmagupta’s method that finds a solution of $x^2 - Ny^2 = 1$ for any nonsquare natural number N. He illustrated it with the case N = 61, for which the least nontrivial solution is extraordinarily large.