The chakravala method (Sanskrit: चक्रवाल विधि) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation.
[3] Jayadeva pointed out that Brahmagupta's approach to solving equations of this type could be generalized, and he then described this general method, which was later refined by Bhāskara II in his Bijaganita treatise.
He called it the Chakravala method: chakra meaning "wheel" in Sanskrit, a reference to the cyclic nature of the algorithm.
Selenius held that no European performances at the time of Bhāskara, nor much later, exceeded its marvellous height of mathematical complexity.
As per popular legend, Chakravala indicates a mythical range of mountains which orbits around the Earth like a wall and not limited by light and darkness.
[6] 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 and Bhaskara offered the first complete solution to the equation, using the chakravala method to find for
the solution This case was notorious for its difficulty, and was first solved in Europe by Brouncker in 1657–58 in response to a challenge by Fermat, using continued fractions.
A method for the general problem was first completely described rigorously by Lagrange in 1766.
[7] Lagrange's method, however, requires the calculation of 21 successive convergents of the simple continued fraction for the square root of 61, while the chakravala method is much simpler.
Selenius, in his assessment of the chakravala method, states Hermann Hankel calls the chakravala method From Brahmagupta's identity, we observe that for given N, For the equation
, this can be scaled down by k (this is Bhaskara's lemma): Since the signs inside the squares do not matter, the following substitutions are possible: When a positive integer m is chosen so that (a + bm)/k is an integer, so are the other two numbers in the triple.
Among such m, the method chooses one that minimizes the absolute value of m2 − N and hence that of (m2 − N)/k.
Then the substitution relations are applied for m equal to the chosen value.
This method always terminates with a solution (proved by Lagrange in 1768).
[9] Optionally, we can stop when k is ±1, ±2, or ±4, as Brahmagupta's approach gives a solution for those cases.
In AD 628, Brahmagupta discovered a general way to find
Finally, from the earlier equations, compose the triples
is useful to find a solution to Pell's Equation, but it is not always the smallest integer pair.
The n = 61 case (determining an integer solution satisfying
), issued as a challenge by Fermat many centuries later, was given by Bhaskara as an example.
, which is scaled down (or Bhaskara's lemma is directly used) to get: For 3 to divide
Now that k is −4, we can use Brahmagupta's idea: it can be scaled down to the rational solution
At each step, we find an m > 0 such that k divides a + bm, and |m2 − 67| is minimal.
That is, we have the new solution: At this point, one round of the cyclic algorithm is complete.
At this point, we could continue with the cyclic method (and it would end, after seven iterations), but since the right-hand side is among ±1, ±2, ±4, we can also use Brahmagupta's observation directly.
Composing the triple (221, 27, −2) with itself, we get that is, we have the integer solution: This equation approximates
"The process of reasoning called "Mathematical Induction" has had several independent origins.
It has been traced back to the Swiss Jakob (James) Bernoulli, the Frenchman B. Pascal and P. Fermat, and the Italian F. Maurolycus.
[...] By reading a little between the lines one can find traces of mathematical induction still earlier, in the writings of the Hindus and the Greeks, as, for instance, in the "cyclic method" of Bhaskara, and in Euclid's proof that the number of primes is infinite."