Editing Brahmagupta–Fibonacci identity (section)

== Application to Pell's equation ==
In its original context, Brahmagupta applied his discovery of this identity to the solution of [[Pell's equation]] ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''Ay''<sup>2</sup>&nbsp;=&nbsp;1. Using the identity in the more general form

:<math>(x_1^2 - Ay_1^2)(x_2^2 - Ay_2^2) = (x_1x_2 + Ay_1y_2)^2 - A(x_1y_2 + x_2y_1)^2, </math>

he was able to "compose" triples (''x''<sub>1</sub>,&nbsp;''y''<sub>1</sub>,&nbsp;''k''<sub>1</sub>) and (''x''<sub>2</sub>,&nbsp;''y''<sub>2</sub>,&nbsp;''k''<sub>2</sub>) that were solutions of ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''Ay''<sup>2</sup>&nbsp;=&nbsp;''k'', to generate the new triple

:<math>(x_1x_2 + Ay_1y_2 \,,\, x_1y_2 + x_2y_1 \,,\, k_1k_2).</math>

Not only did this give a way to generate infinitely many solutions to ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''Ay''<sup>2</sup>&nbsp;=&nbsp;1 starting with one solution, but also, by dividing such a composition by ''k''<sub>1</sub>''k''<sub>2</sub>, integer or "nearly integer" solutions could often be obtained. The general method for solving the Pell equation given by [[Bhaskara II]] in 1150, namely the [[chakravala method|chakravala (cyclic) method]], was also based on this identity.<ref name=stillwell>{{Harvnb|Stillwell|2002|pp=72–76}}</ref>