Talk:Chakravala method

Details

Would be great if someone could add some actual details about how this method works - especially as the article claims it is very simple. Gandalf61 08:16, 14 April 2006 (UTC)

I just added an "example", except that it's way too long, and I gave up before the answer was actually reached, because it's just so tedious. (Simple, but tedious.) If someone adds a proof that this method actually works, without any ad-hoc final steps or handwaving, I might add an implementation in C; but if we can't prove that it will terminate, there's not much point in pursuing that route. --Quuxplusone 22:48, 4 December 2006 (UTC)

It looks as if there is a little "bait and switch" between 61 and 67, as well as being unclear whether the 61 case was solved in the 7th, 9th or 12th centuries. --Rumping (talk) 15:03, 14 August 2008 (UTC)

I just finished the example, which required only two more iterations. In maths, persistence is your friend! Xanthoxyl (talk) 09:50, 29 January 2009 (UTC)

Nearest-square continued fraction

I'm an algorithmist, not a number-theoretician, but most NT's I've spoken to agree with me that the chakravala is easiest viewed (and thus implemented) as a special type of CF (continued fraction).

The classical CF method (aka "English method") of solving Pell's equation involves an RCF (Regular Continued Fraction). The chakravala corresponds (in CF terms) to the NSCF (Nearest-Square Continued Fraction). An Indian mathematician named A.A.K. Ayyangar noticed this in the 1930's, but his work remains relatively unknown (Selenius cites AAK's results but nevertheless manages to claim credit for himself for unravelling the "true nature of chakravala"). If you have an algorithm for RCF it is easily adapted to yield NSCF, and also to get another variant called NICF (Nearest-Integer CF). Both NSCF and NICF are optimal CF's (ie: are the shortest-length CF's possible).

A friend of mine hosts a Number Theory website and we've co-written some papers on this. I'll come back with links when I can find them. DeadHead52 (talk) 11:32, 12 September 2008 (UTC)

Confusing sentence

What does this mean? "The problems Brahmagupta, in 628, used the chakravala method to solve indeterminate quadratic equations, including the Pell equation" Shreevatsa (talk) 13:43, 29 January 2009 (UTC)

I've revised it. Xanthoxyl (talk) 15:01, 29 January 2009 (UTC)

Still doesn't make sense -- how could Brahmagupta use the chakravala method in 628 when the article says it's due to Jayadeva and Bhaskara in the 10th and 12th centuries? Shreevatsa (talk) 15:38, 29 January 2009 (UTC)

Because Brahmagupta came up with the rough idea but could not generalize it effectively. Xanthoxyl (talk) 01:47, 30 January 2009 (UTC)

Thanks, makes sense now :) I guess the article should either mention something like "the chakravala method has its roots in Brahmagupta's..." or say "...solved by Brahmagupta in 628 using the idea that led to the chakravala method were..." or something like that, to avoid the same confusion for others. Shreevatsa (talk) 02:06, 30 January 2009 (UTC)

The example

There is something wrong with the example. In the very first iteration, we have "and take t so that the absolute value of m^2 − 67 is minimized. The result is t = − 2, m = 7, m^2 − 67 = − 18." But if we had taken t = 3 instead, then m = -8 and m^2 - 67 = -3, whose abs value is lower than -18. 91.189.72.18 (talk) 18:43, 21 February 2009 (UTC)

Hmm, that seems right! The article needs to be fixed... How did you happen to catch it? Shreevatsa (talk) 19:41, 21 February 2009 (UTC)

I've added an explanation. Xanthoxyl (talk) 19:53, 21 February 2009 (UTC)

Makes sense now. Perhaps we should include a general description of the method before the examples, including details like this. Shreevatsa (talk) 20:03, 21 February 2009 (UTC)

There's a similar problem with the second iteration t = -2 would produce a smaller value. The first exception (new x would be 0) doesn't apply. Why is it discarded? —Preceding unsigned comment added by 64.213.65.34 (talk) 09:31, 1 April 2009 (UTC) http://cs.annauniv.edu/insight/insight/maths/algebra/indet/chakra.htm shows a method that actually works —Preceding unsigned comment added by 64.213.65.34 (talk) 10:28, 1 April 2009 (UTC)

It looks like t is also being chosen such that $m>0$. This makes sense, since we want x and y to be increasing, so I've added a small change to point this out, but this should probably be explained better. It's not explicitly mentioned in the version you linked, either, but it seems it is also necessary that $p_{r}>0$ when following that version. Asztal (talk) 18:31, 4 May 2009 (UTC)

Question

How did Brahmagupta use this method to get the answer for N=61, because this method can be used to find other solutions for the equation Nx^2 + k = y^2 if and only if one set of value for x and y are known? The solution set for N=61 is the first set of solution using which other set of solutions can be found. So this simply means that Brahmagupta had used other method to solve. Can anyone explain? Ranjitr303 (talk) 05:26, 25 June 2010 (UTC)

No, the method can be used to solve Nx² + 1 = y² starting from a solution to Nx² + k = y² for any k. Thus for N=61, we have a solution (61)1² + 3 = 8², i.e. a solution x=1, y=8, k=3. Now iterate, etc. BTW, Brahmagupta did not solve the N=61 case; the article states it was solved by Jayadeva and Bhaskara II. Shreevatsa (talk) 21:24, 25 June 2010 (UTC)

The cases where k is up to sign 1,2,4

The sentences " Optionally, we can stop when k is ±1, ±2, or ±4, as Brahmagupta's approach gives a solution for those cases." and "but since the right-hand side is among ±1, ±2, ±4, we can also use Brahmagupta's observation directly." are bit unclear to me. What exactly is the observation, and how does one proceed? Evilbu (talk) 13:39, 13 August 2012 (UTC)