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The '''''chakravala'' method''' ({{lang-sa|चक्रवाल विधि}}) is a cyclic [[algorithm]] to solve [[Indeterminate equation|indeterminate]] [[quadratic equation]]s, including [[Pell's equation]]. It is commonly attributed to [[Bhāskara II]], (c. 1114 – 1185 CE)<ref name=SBI200> Hoiberg & Ramchandani – Students' Britannica India: Bhaskaracharya II, page 200</ref><ref name=Kumar23>Kumar, page 23</ref> although some attribute it to [[Jayadeva (mathematician)|Jayadeva]] (c. 950 ~ 1000 CE).<ref name=Plofker474>Plofker, page 474</ref> Jayadeva pointed out that [[Brahmagupta]]'s approach to solving equations of this type could be generalized, and he then described this general method, which was later refined by Bhāskara II in his ''[[Bijaganita]]'' treatise. He called it the Chakravala method: ''chakra'' meaning "wheel" in [[Sanskrit]], a reference to the cyclic nature of the algorithm.<ref name= Goonatilake127>Goonatilake, page 127 &ndash; 128</ref> E. O. Selenius held that no European performances at the time of Bhāskara, nor much later, exceeded its marvellous height of mathematical complexity.<ref name=SBI200/><ref name= Goonatilake127/>
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This method is also known as the '''cyclic method''' and contains traces of [[mathematical induction]].<ref>Cajori (1918), p. 197<blockquote>"The process of reasoning called "Mathematical Induction" has had several independent origins. It has been traced back to the Swiss Jakob (James) Bernoulli, the Frenchman B. Pascal and P. Fermat, and the Italian F. Maurolycus. [...] By reading a little between the lines one can find traces of mathematical induction still earlier, in the writings of the Hindus and the Greeks, as, for instance, in the "cyclic method" of Bhaskara, and in Euclid's proof that the number of primes is infinite."</blockquote></ref>
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[[Brahmagupta]] in 628 CE studied indeterminate quadratic equations, including [[Pell's equation]]
 
:<math>\,x^2 = Ny^2 + 1,</math>
 
for minimum integers ''x'' and ''y''. Brahmagupta could solve it for several ''N'', but not all.
 
Jayadeva (9th century) and Bhaskara (12th century) offered the first complete solution to the equation, using the ''chakravala'' method to find (for the notorious ''N''&nbsp;=&nbsp;61 case)
 
:<math>\,x = 1 766 319 049</math> and <math>\,y = 226 153 980.</math>
 
This case was first solved in [[Europe]] by [[William Brouncker, 2nd Viscount Brouncker|Brouncker]] in 1657–58 in response to a challenge by [[Pierre de Fermat|Fermat]], and a method first completely described by [[Lagrange]] in 1766.<ref>{{MacTutor|class=HistTopics|id=Pell|title=Pell's equation}}</ref> Lagrange's method, however, requires the calculation of 21 successive convergents of the [[continued fraction]] for the [[square root]] of 61, while the ''chakravala'' method is much simpler. <!-- Clas-Olaf? -->Selenius, in his assessment of the ''chakravala'' method, states
 
:"The method represents a best approximation algorithm of minimal length that, owing to several minimization properties, with minimal effort and avoiding large numbers automatically produces the best solutions to the equation. The ''chakravala'' method anticipated the European methods by more than a thousand years. But no European performances in the whole field of [[algebra]] at a time much later than Bhaskara's, nay nearly equal up to our times, equalled the marvellous complexity and ingenuity of ''chakravala''."<ref name=SBI200/><ref name= Goonatilake127/>
 
[[Hermann Hankel]] calls the ''chakravala'' method
:"the finest thing achieved in the theory of numbers before Lagrange."<ref>Kaye (1919), p. 337.</ref>
 
==The method==
The chakravala method for solving Pell's equation is based on the observation by Brahmagupta (see [[Brahmagupta's identity]]) that
 
:<math>(x_1^2 - Ny_1^2)(x_2^2 - Ny_2^2) = (x_1x_2 + Ny_1y_2)^2 - N(x_1y_2 + x_2y_1)^2</math>
 
This defines a "composition" (''samāsa'') of two triples <math>(x_1, y_1, k_1)</math> and <math>(x_2, y_2, k_2)</math> that are solutions of <math>x^2 - Ny^2 = k</math>, to generate the new triple
 
:<math>(x_1x_2 + Ny_1y_2 \,,\, x_1y_2 + x_2y_1 \,,\, k_1k_2).</math>
 
In the general method, the main idea is that any triple <math>(a,b,k)</math> (that is, one which satisfies <math>a^2 - Nb^2 = k</math>) can be composed with the trivial triple <math>(m, 1, m^2 - N)</math> to get the new triple <math>(am + Nb, a+bm, k(m^2-N))</math> for any ''m''. Assuming we started with a triple for which <math>\gcd(a,b)=1</math>, this can be scaled down by ''k'' (this is [[Bhaskara's lemma]]):
 
:<math>a^2 - Nb^2 = k \implies \left(\frac{am+Nb}{k}\right)^2 - N\left(\frac{a+bm}{k}\right)^2 = \frac{m^2-N}{k},</math>
or, since the signs inside the squares do not matter,
:<math>\left(\frac{am+Nb}{|k|}\right)^2 - N\left(\frac{a+bm}{|k|}\right)^2 = \frac{m^2-N}{k}.</math>
 
When a positive integer ''m'' is chosen so that (''a''&nbsp;+&nbsp;''bm'')/''k'' is an integer, so are the other two numbers in the triple. Among such ''m'', the method chooses one that minimizes the absolute value of ''m''<sup>2</sup>&nbsp;&minus;&nbsp;''N'' and hence that of (''m''<sup>2</sup>&nbsp;&minus;&nbsp;''N'')/''k''. Then, (''a'',&nbsp;''b'',&nbsp;''k'') is replaced with the new triple given by the above equation, and the process is repeated. This method always terminates with a solution (proved by Lagrange in 1768).<ref name=stillwell>{{citation | year=2002 | title = Mathematics and its history | author1=[[John Stillwell]] | edition=2 | publisher=Springer | isbn=978-0-387-95336-6 | pages=72–76 | url=http://books.google.com/books?id=WNjRrqTm62QC&pg=PA72}}</ref>
Optionally, we can stop when ''k'' is ±1, ±2, or ±4, as Brahmagupta's approach gives a solution for those cases.
 
==Examples==
===''n'' = 61===
The ''n''&nbsp;=&nbsp;61 case (determining an integer solution satisfying <math>a^2 - 61b^2 = 1</math>), issued as a challenge by Fermat many centuries later, was given by Bhaskara as an example.<ref name=stillwell/>
 
We start with a solution <math>a^2 - 61b^2 = k</math> for any ''k'' found by any means. In this case we can let ''b'' be 1, thus, since <math>8^2 - 61\cdot1^2 = 3</math>, we have the triple <math>(a,b,k) = (8, 1, 3)</math>. Composing it with <math>(m, 1, m^2-61)</math> gives the triple <math>(8m+61, 8+m, 3(m^2-61))</math>, which is scaled down (or [[Bhaskara's lemma]] is directly used) to get:
: <math>\left( \frac{8m+61}{3}, \frac{8+m}{3}, \frac{m^2-61}{3} \right).</math>
 
For 3 to divide <math>8+m</math> and <math>|m^2-61|</math> to be minimal, we choose <math>m=7</math>, so that we have the triple <math>(39, 5, -4)</math>. Now that ''k'' is &minus;4, we can use Brahmagupta's idea: it can be scaled down to the rational solution <math>(39/2, 5/2, -1)\,</math>, which composed with itself three times, with <math>m={7,11,9}</math> respectively, when k becomes square and scaling can be applied, this gives <math>(1523/2, 195/2, 1)\,</math>. Finally, such procedure can be repeated until the solution is found (requiring 9 additional self-compositions and 4 additional square-scalings): <math>(1766319049,\, 226153980,\, 1)</math>. This is the minimal integer solution.
 
===''n'' = 67===
Suppose we are to solve <math>x^2 - 67y^2 = 1</math> for ''x'' and ''y''.<ref>The example in this section is given (with notation <math>Q_n</math> for ''k'', <math>P_n</math> for ''m'', etc.) in: {{citation | year=2009 | title = Solving the Pell equation | author1=Michael J. Jacobson | author2=Hugh C. Williams | publisher=Springer | isbn=978-0-387-84922-5 | page=31 | url=http://books.google.com/books?id=2INzqrEUGzAC&pg=PA31}}</ref>
 
We start with a solution <math>a^2 - 67b^2 = k</math> for any ''k'' found by any means; in this case we can let ''b'' be 1, thus producing <math>8^2 - 67\cdot1^2 = -3</math>. At each step, we find an ''m''&nbsp;>&nbsp;0 such that ''k'' divides ''a''&nbsp;+&nbsp;''bm'', and |''m''<sup>2</sup>&nbsp;&minus;&nbsp;67| is minimal. We then update ''a'', ''b'', and ''k'' to <math>\frac{am+Nb}{|k|}, \frac{a+bm}{|k|}, \text{ and }\frac{m^2-N}{k}</math> respectively.
 
;First iteration
We have <math>(a,b,k) = (8,1,-3)</math>. We want a positive integer ''m'' such that ''k'' divides ''a''&nbsp;+&nbsp;''bm'', i.e. 3 divides 8 + m, and |''m''<sup>2</sup>&nbsp;&minus;&nbsp;67| is minimal. The first condition implies that ''m'' is of the form 3''t'' + 1 (i.e. 1, 4, 7, 10,… etc.), and among such ''m'', the minimal value is attained for ''m'' = 7. Replacing (''a'',&nbsp;''b'',&nbsp;''k'') with <math>\left(\frac{am+Nb}{|k|}, \frac{a+bm}{|k|}, \frac{m^2-N}{k}\right)</math>, we get the new values <math>a = (8\cdot7+67\cdot1)/3 = 41, b = (8 + 1\cdot7)/3 = 5, k = (7^2-67)/(-3) = 6</math>. That is, we have the new solution:
: <math>41^2 - 67\cdot(5)^2 = 6.</math>
 
At this point, one round of the cyclic algorithm is complete.
 
;Second iteration
We now repeat the process. We have <math>(a,b,k) = (41,5,6)</math>. We want an ''m''&nbsp;>&nbsp;0 such that ''k'' divides ''a''&nbsp;+&nbsp;''bm'', i.e. 6 divides 41&nbsp;+&nbsp;5''m'', and |''m''<sup>2</sup>&nbsp;&minus;&nbsp;67| is minimal. The first condition implies that ''m'' is of the form 6''t''&nbsp;+&nbsp;5 (i.e. 5, 11, 17,… etc.), and among such ''m'', |''m''<sup>2</sup>&nbsp;&minus;&nbsp;67| is minimal for ''m''&nbsp;=&nbsp;5. This leads to the new solution ''a''&nbsp;=&nbsp;(41⋅5&nbsp;+&nbsp;67⋅5)/6, etc.:
 
:<math>90^2 - 67 \cdot 11^2 = -7.</math>
 
;Third iteration
For 7 to divide 90 + 11''m'', we must have ''m'' = 2&nbsp;+&nbsp;7''t'' (i.e. 2, 9, 16,… etc.) and among such ''m'', we pick ''m'' = 9.
 
:<math>221^2 - 67\cdot 27^2 = -2.</math>
 
;Final solution
At this point, we could continue with the cyclic method (and it would end, after seven iterations), but since the right-hand side is among ±1, ±2, ±4, we can also use Brahmagupta's observation directly. Composing the triple (221, 27, &minus;2) with itself, we get
:<math> \left(\frac{221^2 + 67\cdot27^2}{2}\right)^2 - 67\cdot(221\cdot27)^2 = 1,</math>
 
that is, we have the integer solution:
:<math> 48842^2 - 67 \cdot 5967^2 = 1.</math>
 
This equation approximates <math>\sqrt{67}</math> (as 48842/5967) to within a margin of about 2 × 10<sup>-9</sup>.
 
==Notes==
{{reflist}}
 
==References==
*[[Florian Cajori]] (1918), Origin of the Name "Mathematical Induction", ''[[The American Mathematical Monthly]]'' '''25''' (5), p. 197-201.
*George Gheverghese Joseph, ''The Crest of the Peacock: Non-European Roots of Mathematics'' (1975).
*G. R. Kaye, "Indian Mathematics", ''Isis'' '''2''':2 (1919), p. 326–356.
*C. O. Selenius, "Rationale of the chakravala process of Jayadeva and Bhaskara II", ''Historia Mathematica'' '''2''' (1975), pp. 167-184.
*C. O. Selenius, "Kettenbruch theoretische Erklarung der zyklischen Methode zur Losung der Bhaskara-Pell-Gleichung", ''Acta Acad. Abo. Math. Phys.'' '''23''' (10) (1963).
*Hoiberg, Dale & Ramchandani, Indu (2000). ''Students' Britannica India''. Mumbai: Popular Prakashan. ISBN 0-85229-760-2
*Goonatilake, Susantha (1998). ''Toward a Global Science: Mining Civilizational Knowledge''. Indiana: Indiana University Press. ISBN 0-253-33388-1.
*Kumar, Narendra (2004). ''Science in Ancient India''. Delhi: Anmol Publications Pvt Ltd. ISBN 81-261-2056-8
*Ploker, Kim (2007) "Mathematics in India". ''The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook'' New Jersey: Princeton University Press. ISBN 0-691-11485-4
*{{cite book
  | last = Edwards
  | first = Harold
  | title = Fermat's Last Theorem
  | publisher = [[Springer Science+Business Media|Springer]]
  | location = New York
  | date = 1977
  | isbn = 0-387-90230-9}}
 
==External links==
*[http://www-groups.dcs.st-and.ac.uk/~history/Miscellaneous/Pearce/Lectures/Ch8_6.html Introduction to chakravala]
 
{{number theoretic algorithms}}
 
[[Category:Brahmagupta]]
[[Category:Diophantine equations]]
[[Category:Number theoretic algorithms]]
[[Category:Indian mathematics]]

Latest revision as of 22:54, 13 November 2014

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