Simultaneous quadratic equations with few or no solutions

Abstract

In this paper, we completely solve the simultaneous Diophantine equations x 2 − az 2 = 1, y 2 − bz 2 = 1 provided the positive integers a and b satisfy b − a ϵ {1,2,4}. Further, we show that these equations possess at most one solution in positive integers ( x, y, z) if b − a is a prime or prime power, under mild conditions. Our approach is (relatively) elementary in nature and relies upon classical results of Ljunggren on quartic Diophantine equations.