Pell’s equation

Abstract

Number theory is that branch of mathematics that is concerned with the properties of numbers. For this reason, number theory, which has a 4000 years of rich history, has traditionally been considered as pure mathematics. The theory of numbers has always occupied a unique position in the world of mathematics. This is due to unquestionable historical importance of the subject. It is one of the few disciplines having demonstrable results that predate the very idea of a university or an academy. The natural numbers have been known to us for so long that mathematician Leopold Kronecker once remarked, “God created the natural numbers, and all the rest is the work of man”. Far from being a gift from Heaven, number theory has a long and sometimes painful evolution.
The theory of continued fractions begins with Rafael Bombelli, the last of great algebraists of Renaissance Italy. In his L’Algebra Opera (1572) , Bombelli attempted to find square roots by using infinite continued fractions. One of the main uses of continued fraction is to find the approximate values of irrational numbers.
Srinivas Ramanujan has no rival in the history of mathematics. His contribution to number theory is quite significant. G.H.Hardy, commenting on Ramanujan’s work, said “On this side (of Mathematics) most recently I have never met his equal, and I can only compare him with Euler or Jacobi”.
Pell’s equation ,was probably first studied in the case .Early mathematicians, upon discovering that is irrational, realized that although one cannot solve the equation in integers,one can at least solve the “next best things”. The early investigators of Pell equation were the Indian mathematicians
Brahmagupta and Bhaskara. In particular Bhaskara studied Pell’s equation for the values and Bhaskara found the solution for .
Fermat was also interested in the Pell’s equation and worked out some of the basic theories regarding Pell’s equation. It was Lagrange who discovered the complete theory of the equation, .Euler mistakenly named the equation to John Pell. He did so apparently because Pell was instrumental in writing a book containing these equations. Brahmagupta has left us with this intriguing challenge: “A person who can, within a year, solve is a mathematician.” In general Pell’s equation is a Diophantine equation of the form , where is a positive non square integer and has a long fascinating history and its applications are wide and Pell’s equation always has the trivial solution , and has infinite solutions and many problems can be solved using Pell’s equation.