The Method
The chakravala method for solving Pell's equation is based on the observation by Brahmagupta (see Brahmagupta's identity) that
This defines a "composition" (samāsa) of two triples and that are solutions of, to generate the new triple
In the general method, the main idea is that any triple (that is, one which satisfies ) can be composed with the trivial triple to get the new triple for any m. Assuming we started with a triple for which, this can be scaled down by k (this is Bhaskara's lemma):
or, since the signs inside the squares do not matter,
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 m2 − N and hence that of (m2 − N)/k. Then, (a, b, 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). Optionally, we can stop when k is ±1, ±2, or ±4, as Brahmagupta's approach gives a solution for those cases.
Read more about this topic: Chakravala Method
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