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  pr and kr . '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 pr, kr, Xr, Yr, choose pr+1 such that the following two conditions are satisfied:-

(a) pr + pr+1 is divisible by kr and

(b) | pr+12 -D| is minimum .

After choosing pr+1, the parameter kr+1 is calculated as follows :

kr+1= ( (pr+1)2 - D) / kr .

The values for Xr+1 and Yr+1 are given by,

Xr+1 = (pr+1* Xr + D * Yr) /  | kr |

Yr+1 = (pr+1* Yr + Xr) / | kr | .

3) Step 2 is continued with 'r' being incremented in each stage. We stop when ki of a step = 1. The values of  Xi and Yi 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).

 r pr kr Xr Yr 0 1 1 1 0 1 3 1 3 1 2 3 1 17 6 3 3 1 99 35 4 3 1 577 204

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

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

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

Another Example