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Pell's Equation
# Pell's Equation

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## Definition

Pell's equation is defined as follows for natural numbers (x, y, N):

${x}^{2}-N.{y}^{2}=1$

It can be used to approximate irrational square roots by rational numbers.
For more details, see:

- Pell's equation -- Wikipedia
- Pell's equation -- Encyclopedia of Mathematics
- Diophantine equation -- Wikipedia
- Chakravala method -- Wikipedia
- Brahmagupta -- Wikipedia
- Bhaskara II -- Wikipedia
- University of St Andrews, UK
- J.-A. Serret (Ed.), Oeuvres de Lagrange, vol. 1, pp. 671–731, 1867.
- Edward J. Barbeau, Pell's Equation, Springer, 2003.
- André Weil, Number Theory: An approach through history from Hammurapi to Legendre, Birkhaüser, 2007 (reprint of 1984 edition).

## Fundamental Solutions

The fundamental solutions to Pell's equation (also known as "minimal solutions")
presented below were computed in 2012 using a Python program that implements the
chakravala method
(attributed to Bhaskara II).
N is skipped if it is not square-free (i.e., if it is a perfect square or
a multiple of a square).