On the Diophantine equations $x^2-Dy^2=-1$ and $x^2-Dy^2=4$

Abstract

In this paper, using only the St$ \ddot{o} $rmer theorem and its generalizations on Pell's equation and fundamental properties of Lehmer sequence and the associated Lehmer sequence, we discuss the Diophantine equations $x^2-Dy^2 = -1$ and $x^2-Dy^2 = 4$. We obtain the relation between a positive integer solution (x, y) of the Diophantine equation $x^2-Dy^2 = -1$ and its fundamental solution if there is exactly one or two prime divisors of y not dividing D. We also obtain the relation between a positive integer solution (x, y) of the Diophantine equation $x^2-Dy^2 = 4$ and its minimal positive solution if there is exactly two prime divisors of y not dividing D.