F10: **The Pell Equation in India**

**Author**: Keith Jones and Toke Knudson

**Description**:
The Pell equation is the Diophantine equation
x^{2} - Ny^{2} = 1

where N is a non-square, positive integer. The equation has infinitely many solutions in positive integers x and y, though finding a solution is not trivial.

In modern mathematics, the method of solving the Pell equation via continued fractions was developed by Lagrange (1736-1813). However, much earlier, Indian mathematician made significant contributions to the study of the Pell equation and its solution. Brahmagupta (b. 598 CE) discovered that the Pell equation can be solved if a solution to

x^{2} - Ny^{2} = k

where k=-1,2,-2,4,-4 is known. Later a method, a cyclic algorithm known in Sanskrit as cakravala, to solve the Pell equation was developed by Jayadeva and Bhaskara ii (b. 1114 CE).

Although more advanced students will naturally find the algebraic simplifications involved in certain tasks to be more straightforward, the only mathematical content pre-requisites are required in either version is some basic arithmetic and (high school level) algebraic skills. The major distinction between the two versions of this project is instead the degree of general mathematical maturity expected. Both versions include an open-ended ``comparisons and conjectures' ' penultimate section that could be omitted (or expanded upon) depending on the instructor's goals for the course. The student tasks included in other sections of the project are essentially the same in the two versions as well, but differently ordered in a fashion that renders Version M 26.1 somewhat less open-ended than Version M 26.2.

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