Pell's Equation

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pell's equation is any Diophantine equation of the form x^2-ny^2=1, where n is a nonsquare integer and x and y are integers. Trivially, x = 1 and y = 0 always solve this equation. Lagrange proved that for any natural number n that is not a perfect square there are x and y > 0 that satisfy Pell's equation. Moreover, infinitely many such solutions of this equation exist. These solutions yield good rational approximations of the form x/y to the square root of n. The name of this equation arose from Leonhard Euler's mistakenly attributing its study to John Pell. Euler was aware of the work of Lord Brouncker, the first European mathematician to find a general solution of the equation, but apparently confused Brouncker with Pell. This equation was first studied extensively in ancient India, starting with Brahmagupta, who developed the chakravala method to solve Pell's equation and other quadratic indeterminate equations in his Brahma Sphuta Siddhanta in 628, about a thousand years before Pell's time.

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