Chakravala Method - History

History

Brahmagupta in 628 CE studied indeterminate quadratic equations, including Pell's equation

for minimum integers x and y. Brahmagupta could solve it for several N, but not all.

Jayadeva (9th century) and Bhaskara (12th century) offered the first complete solution to the equation, using the chakravala method to find (for the notorious N = 61 case)

and

This case was first solved in Europe by Brouncker in 1657–58 in response to a challenge by Fermat, and a method first completely described by Lagrange in 1766. Lagrange's method, however, requires the calculation of 21 successive convergents of the continued fraction for the square root of 61, while the chakravala method is much simpler. Selenius, in his assessment of the chakravala method, states

"The method represents a best approximation algorithm of minimal length that, owing to several minimization properties, with minimal effort and avoiding large numbers automatically produces the best solutions to the equation. The chakravala method anticipated the European methods by more than a thousand years. But no European performances in the whole field of algebra at a time much later than Bhaskara's, nay nearly equal up to our times, equalled the marvellous complexity and ingenuity of chakravala."

Hermann Hankel calls the chakravala method

"the finest thing achieved in the theory of numbers before Lagrange."