Simultaneous Pell Equations x^2-my^2= 1 and y^2-5z^2 = 1

Abstract

This paper will discuss the solutions on the simultaneous Pell equations x^2-my^2= 1 and y^2-5z^2 = 1 where ݉ is any positive integer that is not a perfect square and (݉m, 5) = 1. By finding the fundamental solutions of y^2-5z^2 = 1, the possibility of the parity x, y and ݉ will be obtained. Then, the lemmas and theorems will be developed. The solutions to these equations are (2^gs+1,9,4,2^a,k) and (x,y,z,m) = (݉x_i,9,4,2,k_i+1) for certain positive integers a, g, s, k and ݅i ∈ ܰN.