Pell's equation is a mathematical equation of the form x2 - Dy2 = 1, where D is a non-square positive integer. It is named after John Pell, who studied the equation in the 17th century.
There are several resources available for solving Pell's equation, including textbooks, online resources, and mathematical software programs. Some popular resources include "An Introduction to the Theory of Numbers" by Ivan Niven, "Solving the Pell Equation" by Michael Jacobson Jr., and the software program Wolfram Mathematica.
Pell's equation has a close relationship with the theory of continued fractions. In particular, the fundamental solution to Pell's equation (the smallest positive integer solution) can be found using the continued fraction expansion of the square root of D. This connection has been utilized in various algorithms for solving Pell's equation.
While Pell's equation may seem like a purely theoretical concept, it has several real-world applications. It has been used in cryptography, specifically in the development of RSA encryption. It is also used in the study of Diophantine equations, which have applications in fields such as number theory and cryptography.
There is no general method for solving Pell's equation for all values of D. However, there are several algorithms that can be used to find solutions for specific values of D. These include the Chakravala method, the Lagrange method, and the continued fraction method. It is also possible to use computer algorithms to find solutions for large values of D.