__CHAKRAVALA TECHNIQUE__

Chakravala is a cyclic process,
developed by the Indian mathematicians. Given a solution for Vargaprakrtis of the form X^{2}-
DY^{2}=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 p_{0}=1 and k_{0}=1. This is the case of
getting a trivial solution i.e., X_{0}=1,Y_{0}=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+1}^{2}-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 X^{2}
- 8Y^{2}=1.

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

r |
p |
k |
X |
Y |

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 p_{r+1}
for r = 1. * p _{r+1}^{2} 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 X