CHAKRAVALA TECHNIQUE

Chakravala is a cyclic process, developed by the Indian mathematicians. Given a solution for Vargaprakrtis of the form X²- DY²=1 (even trivial one like x=1, y=0!), this technique can be used to find further integeral solutions. The discoverer of the method is not known but there is a reference to Chakravala in a commentary "Sundari" by Udayadivakara on Laghubhaskariya by Bhaskaracharya.

In each step of the process, we use two parameters  p_r and k_r . 'r' is the number of the current step, starting from 0. The algorithm goes as follows.

1) Initially (for r=0), take p₀=1 and k₀=1. This is the case of getting a trivial solution i.e., X₀=1,Y₀=0.

2)When r >0, given p_r, k_r, X_r, Y_r, choose p_r+1 such that the following two conditions are satisfied:-

(a) p_r + p_r+1 is divisible by k_r and

(b) | p_r+1² -D| is minimum .

After choosing p_r+1, the parameter k_r+1 is calculated as follows :

k_r+1= ( (p_r+1)² - D) / k_{r .}

The values for X_r+1 and Y_r+1 are given by,

X_r+1 = (p_r+1* X_r + D * Y_r) /  | k_r |

Y_r+1 = (p_r+1* Y_r + X_r) / | k_r | .

3) Step 2 is continued with 'r' being incremented in each stage. We stop when k_i of a step = 1. The values of  X_i and Y_i at this stage form another solution to the given equation.

Now, let us consider the Vargaprakrti X² - 8Y²=1.

Let us start finding non-trivial solutions for the above equation from the trivial solution (1,0).

Note, how we are choosing  p_r+1 for r = 1.  p_r+1² has to be the nearest square to D so that p_r + p_r+1 is divisible by k_r .

Here D = 8. The nearest square to 8 is 9. Hence,  p_r+1= 3.  p_r + p_r+1= 1 + 3 = 4, which is also divisible by 1. 

Then we calculate k_r+1, X_r+1, Y_r+1, using the formulas stated above. Note that, for each ' r ', we get k_r+1 to be 1. Hence, all the corresponding values of X_r and Y_r  form a solution. But this will not be the case for all Vargaprakrtis. Try to  solve the equation X² - 61Y²=1.

Another Example