Malaysian Journal of Mathematical Sciences, April 2017, Vol. 11(S)
Special Issue: The 2nd International Conference and Workshop on Mathematical Analysis (ICWOMA 2016)
Simultaneous Pell Equations \(x^2-my^2=1\) and \(y^2-pz^2=1\)
Sihabudin, N. A., Sapar, S. H., and Johari, M. A. M.
Corresponding Email: [email protected]
Received date: -
Accepted date: -
Abstract:
Pell equation is a special type of Diophantine equations of the form \(x^2-my^2=1\), where \(m\) is a positive non-square integer. Since \(m\) is not a perfect square, then there exist infinitely many integer solutions \((x, y)\) to the Pell equation. This paper will discuss the integral solutions to the simultaneous Pell equations \(x^2-my^2=1\) and \(y^2-pz^2=1\), where \(m\) is square free integer and \(p\) is odd prime. The solutions of these simultaneous equations are of the form of \((x,y,z,m)=(y_{n}^{2}t \pm 1, y_n, z_n, y_{n}^{2}t^{2} \pm 2t)\) and \((\frac{y_{n}^{2}}{2}t \pm 1, y_n, z_n, \frac{y_{n}^{2}}{4}t^{2} \pm t)\) for \(y_n\) odd and even respectively, where \(t \in \mathbb{N}\).
Keywords: Simultaneous Pell equations, Pell equation and parity
