Brahmagupta (598–.670 CE),Brahmagupta went on to give a recurrence relation for generating solutions to certain instances of Diophantine equations of the second degree such as (called Pell’s Equation) by using the Euclidean algorithm. The Euclidean algorithm was known to him as the “pulverizer” since it breaks numbers down into ever smaller pieces.
The nature of squares:
18.64. [Put down] twice the square-root of a given square by a multiplier and increased or diminished by an arbitrary [number]. The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed.
18.65. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rupas.
The key to his solution was the identity,
which is a generalization of an identity that was discovered by Diophantus,
Using his identity and the fact that if and are solutions to the equations and , respectively, then is a solution to , he was able to find integral solutions to the Pell’s equation through a series of equations of the form . Unfortunately, Brahmagupta was not able to apply his solution uniformly for all possible values of N, rather he was only able to show that if has an integer solution for k = ±1, ±2, or ±4, then has a solution. The solution of the general Pell’s equation would have to wait for Bhaskara II in c. 1150 CE.
to be continue ……on http://www.speak2world.wordpress.com