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

Abstract

In this paper, using only the Störmer 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.

Keywords

• Diophantine equations
• Pell equations
• minimal solutions
• Lehmer sequences