On the solutions of Pellian equation U^2-DV^2=K^2 N

Abstract

In this paper we consider a class of Pell’s equation
U^2-DV^2=k^2 N, (1)
where D,N are positive integers, D is non-square and k is any integer. When (u,v) satisfy (1), we define (u+v√D)/k to be the solution of (1). We first introduce the class of solutions of (1) and call (u+v√D)/k to be the fundamental solution of the class, if v is the smallest positive value which occurs in the solutions of that class.
We first derive the necessary and sufficient condition for any two solutions of (1) to belong to the same class and the bounds for the values of u,v occurring in the fundamental solution. We also derive an explicit formula which gives all the solutions of (1). We further present some interesting recurrence relations connecting the values of u,v. Finally, we obtain the results for total number of positive solutions of (1) not exceeding any given positive real number Z.