Some Results on Pell’s Equation

An English mathematician John Pell (1611-1685), who taught mathematics in Holland, at the universities of msterdam and reda in 1640‘s had introducing a newly type Diophantine equation which latterly famous as Pell‘s equation. It has a long attractive history, [4]. One of the main reasons for the popularity of Pell‘s equation is the fact that many natural questions that one might ask about integer‘s leads to a quadratic equation in two variables, which can be casted as a Pell‘s equation,[1]. Fermat was also paying attention in the Pell‘s equation and has given ideas on some of the basic theories regarding Pell‘s equation. It was Lagrange, who discovered the complete theory of the equation

, [5].

Euler erroneously named the equation to John Pell. He did so apparently because Pell was influential in writing a book containing these equations. World fame Indian astronomer and mathematician Brahmagupta has left us with this intriguing challenge, ― person who can, within a year, solve is a mathematician.‖[2]. In general Pell‘s equation is a iophantine 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 (x, y) = (1,0), and has infinite solutions and many problems can be solved using Pell‘s equation,[3].

The Pell equation is a Diophantine equation of the form . We want to find all integer pairs (x,y) that satisfy the equation. Since any given solution (x,y) yields multiple solution we restrict ourselves to those solutions where x and y are positive integers. We generally take to be a positive non-square integer; otherwise there are only uninteresting solutions. If then in the case and or in the case , if , then (y arbitrary) and if d is non-zero square, then dy and x are consecutive squares, implying that . One of the main property of the Pell equation has always the trivial solution . The following result is well known If is the convergent to the irrational number x, then Theorem 1.1. If is a convergent of the continued fraction expansion of then (p,q) is a

Proof. If is a convergent of , then And, therefore Now These two inequalities combine to yield Example 1.1. Let . Then The first two convergent of are Now calculating, , we find that Hence provides a positive solution of Corollary: If is a positive integer and it is not a perfect square, then the continued fraction expansion of necessarily has the from Theorem 1.2. Let be the fundamental solution of . And every pair of integers defined by the following condition. Then this will be also a positive solution, where n=1,2,3,…. Proof. Further, because and are positive and are both positive integers. Since , is a solution of , we have Hence, is a solution. Theorem 1.3. If and and , then and for positive . Proof. We will prove this by induction, observe that and Since , and is a positive integer it is clear that . So, the result holds for . Now assume the solution with and positive integers greater than 1. We have, Therefore, and . We know that, and , so also and implies that . Therefore, we have and .

Hence we have seen simple method for constructing some results and relation between trivial solutions of the Pell‘s equation under some certain conditions with its corresponding complement of solution which give rise to new idea about solution pairs.

1. Alter, R. and Curtz, T.B. (1974): A note on congruent numbers. Math. Comp., 303–305. Nagell, T. (1951): Introduction to Number Theory, New York: Wiley. 2. Burton, D.M. (2007): Elementary number theory, 6th ed., McGraw Hill, New York. 3. Dickson, L.E. (1999): History of the Theory of Numbers, AMS Chelsea, Vol. II (ISBN 0-8218-1935-6). 4. Hardy, G.H. and Wright, E.M. (1979): An Introduction to the Theory of Numbers, 5th ed., Oxford University Press, New York. 5. Stephens, N.M. (1975): Congruence properties of congruent numbers, Bull. London Math. Soc., 7: pp. 182-184.

Research Scholar, VKSU, Ara, Bihar