Talk:Chakravala method/Archive 1
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| Archive 1 |
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)
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)
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)
- 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)
- 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)