Question about the chakravala method on solving Pell's equation

Question

I am currently reading on this old way of Pell's equation: http://en.wikipedia.org/wiki/Chakravala_method

Looking at the section where they consider $N = 61$, it is not clear to me if the solution using this method always will result in the minimal solution for (x,y)...? As far as I understand, Bhramagupta's identity does not imply that you can break down a random solution into a minimal solution.

Thanks.

Answer 1

I have no idea about this particular method. However, Lagrange and Gauss eventually settled on a method using binary quadratic forms that solves Pell's equation, and does other things, using only integer operations... Here is output from my program; a very short description of the method is in http://math.blogoverflow.com/2014/08/23/binary-quadratic-forms-over-the-rational-integers-and-class-numbers-of-quadratic-%EF%AC%81elds/

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 
    jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./Pell 61


0  form   1 14 -12   delta  -1
1  form   -12 10 3   delta  4
2  form   3 14 -4   delta  -3
3  form   -4 10 9   delta  1
4  form   9 8 -5   delta  -2
5  form   -5 12 5   delta  2
6  form   5 8 -9   delta  -1
7  form   -9 10 4   delta  3
8  form   4 14 -3   delta  -4
9  form   -3 10 12   delta  1
10  form   12 14 -1   delta  -14
11  form   -1 14 12   delta  1
12  form   12 10 -3   delta  -4
13  form   -3 14 4   delta  3
14  form   4 10 -9   delta  -1
15  form   -9 8 5   delta  2
16  form   5 12 -5   delta  -2
17  form   -5 8 9   delta  1
18  form   9 10 -4   delta  -3
19  form   -4 14 3   delta  4
20  form   3 10 -12   delta  -1
21  form   -12 14 1   delta  14
22  form   1 14 -12

 disc   244
Automorph, written on right of Gram matrix:  
183241189  2713847760
226153980  3349396909


 Pell automorph 
1766319049  13795392780
226153980  1766319049

Pell unit 
1766319049^2 - 61 * 226153980^2 = 1 

=========================================

Pell NEGATIVE 
29718^2 - 61 * 3805^2 = -1 

=========================================

  4 PRIMITIVE 
1523^2 - 61 * 195^2 = 4 

=========================================

  -4 PRIMITIVE 
39^2 - 61 * 5^2 = -4 

=========================================

61       61

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus

.............................................