On the number of solutions of $x^2-4m(m+1)y^2=y^2-bz^2=1$

Abstract

In this paper, using a result of Ljunggren and some results on primitive prime factors of Lucas sequences of the first kind, we prove the following results by an elementary argument: if $m$ and $b$ are positive integers, then the simultaneous Pell equations \[ x^2-4m(m+1)y^2=y^2-bz^2=1\] possesses at most one solution $(x,y,z)$ in positive integers.