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Wikipedia / Chakravala method

The chakravala method () is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE) although some attribute it to Jayadeva (c. 950 ~ 1000 CE). 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. C.-O. Selenius held that no European performances at the time of Bhāskara, nor much later, exceeded its marvellous height of mathematical complexity.

This method is also known as the cyclic method and contains traces of mathematical induction.

History

Chakra in Sanskrit means cycle. As per popular legend, Chakravala indicates a mythical range of mountains which orbits around the Earth like a wall and not limited by light and darkness.

Brahmagupta in 628 CE studied indeterminate quadratic equations, including Pell's equation

for minimum integers x and y. Brahmagupta could solve it for several N, but not all.

Jayadeva and Bhaskara offered the first complete solution to the equation, using the chakravala the solution

This case was notorious for its difficulty, and was first solved in Europe by Brouncker in 1657–58 in response to a challenge by Fermat, using continued fractions. A method for the general problem was first completely described rigorously by Lagrange in 1766. which can be squared to eliminate the remaining 11 iterations) for successive convergents of the simple continued fraction for the square root of 61, while the chakravala method is much simpler. Selenius, in his assessment of the chakravala method, states

  1. "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."

Hermann Hankel calls the chakravala method

  1. "the finest thing achieved in the theory of numbers before Lagrange."

The method

From Brahmagupta's identity, we observe that for given N,

, this allows the "composition" (samāsa into a new triple

for any m, this can be scaled down by k (this is Bhaskara's lemma):

Since the signs inside the squares do not matter, the following substitutions are possible:

When a positive integer m is chosen so that (a+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 m2N and hence that of (m2N)/k. Then the substitution relations are applied for m equal to the chosen value. This results in a new triple (a, b, k is found. This method always terminates with a solution, as proved by Lagrange in 1768. Optionally, we can stop when k is ±1, ±2, or ±4, as Brahmagupta's approach gives a solution for those cases.

Brahmagupta's composition method

, when k is ±1, ±2, or ±4.

k = ±1

Using Brahmagupta's identity with itself:

.

gives a solution:

yields a second:

k = ±2

,

,

k = 4

is even:

If a.

is odd,

k = −4

.

, to get

  1. .

This gives us the solutions

is useful to find a solution to Pell's Equation.

Examples

n = 61

The n=), issued as a challenge by Fermat many centuries later, was given by Bhaskara as an example.

for any k found by any means. In this case we can let b, which is scaled down (or Bhaskara's lemma is directly used) to get:

. Now that k is . This is the minimal integer solution.

n = 67

for x and y.

for any k found by any means; in this case we can let b. At each step, we find an m>0 such that k divides a+bm, and |m267| is minimal. We then update a, b, and k respectively.

  1. First iteration

. We want a positive integer m such that k divides a+bm, i.e. 3 divides 8 + m, and |m267| is minimal. The first condition implies that m is of the form 3t + 1 (i.e. 1, 4, 7, 10,… etc.), and among such m, the minimal value is attained for m = 7. Replacing (a,b,k. That is, we have the new solution:

At this point, one round of the cyclic algorithm is complete.

  1. Second iteration

. We want an m>0 such that k divides a+bm, i.e. 6 divides 41+5m, and |m267| is minimal. The first condition implies that m is of the form 6t+5 (i.e. 5, 11, 17,… etc.), and among such m, |m267| is minimal for m=

  1. Third iteration

For 7 to divide 90 + 11m, we must have m = 2+7t (i.e. 2, 9, 16,… etc.) and among such m, we pick m = 9.

  1. 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, 2) with itself, we get

that is, we have the integer solution:

.

Notes

References

  • Florian Cajori (1918), Origin of the Name "Mathematical Induction", The American Mathematical Monthly 25 (5), pp.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), pp.326–356.

  • Clas-Olaf Selenius, "Rationale of the chakravala process of Jayadeva and Bhaskara II" , Historia Mathematica 2 (1975), pp.167–184.

  • Clas-Olaf Selenius, "Kettenbruchtheoretische Erklärung der zyklischen Methode zur Lösung der Bhaskara-Pell-Gleichung", Acta Acad. Abo. Math. Phys. 23 (10) (1963), pp.1–44.

  • Hoiberg, Dale & Ramchandani, Indu (2000). Students' Britannica India. Mumbai: Popular Prakashan.

  • Goonatilake, Susantha (1998). Toward a Global Science: Mining Civilizational Knowledge. Indiana: Indiana University Press. .

  • Kumar, Narendra (2004). Science in Ancient India. Delhi: Anmol Publications Pvt Ltd.

  • Ploker, Kim (2007) "Mathematics in India". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook New Jersey: Princeton University Press.