On the determination of solutions of simultaneous Pell equations x(2) - (a(2)-1) y(2) = y(2) - pz(2)=1

Abstract

where p is prime and a > 1. Assuming the solutions of the Pell equation x(2) -(a(2) - 1) y(2) = 1 are x = x(m) and y = y(m) with m = 2, we prove that the system (0.1) has solutions only when m = 2 or m = 3. In the case of m = 3, we show that p = 2 and give the solutions of (0.1) in terms of Pell and Pell-Lucas sequences. When m = 2 and p = 3(mod 4), we determine the values of a, x, y, and z. Lastly, we show that (0.1) has no solutions when p = 1(mod 4).