Talk:Chakravala method: Difference between revisions

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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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

@@ Line 10: / Line 10: @@
 :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 [http://www-groups.dcs.st-andrews.ac.uk/~history/Miscellaneous/Pearce/Lectures/Ch8_6.html ad-hoc final steps] or [http://www.math.sfu.ca/histmath/India/12thCenturyAD/Chakravala.html handwaving], I might add an implementation in [[C (programming language)|C]]; but if we can't prove that it will terminate, there's not much point in pursuing that route. --[[User:Quuxplusone|Quuxplusone]] 22:48, 4 December 2006 (UTC)
-::It looks as if the 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. --[[User:Rumping|Rumping]] ([[User talk:Rumping|talk]]) 15:03, 14 August 2008 (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. --[[User:Rumping|Rumping]] ([[User talk:Rumping|talk]]) 15:03, 14 August 2008 (UTC)
-===============
-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).
+:::I just finished the example, which required only two more iterations. In maths, persistence is your friend! [[User:Xanthoxyl|Xanthoxyl]] ([[User talk:Xanthoxyl|talk]]) 09:50, 29 January 2009 (UTC)
-The classical CF method (aka "English method") of solving Pell's equation involves an RCF (''Regular Continued Fraction'').
+==Nearest-square continued fraction==
-The ''chakravala'' corresponds (in CF terms) to the NSCF (''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).
-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).
+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.
-[[User:DeadHead52|DeadHead52]] ([[User talk:DeadHead52|talk]]) 11:32, 12 September 2008 (UTC)
+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. [[User:DeadHead52|DeadHead52]] ([[User talk:DeadHead52|talk]]) 11:32, 12 September 2008 (UTC)
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