Uniqueness of Solutions to Simultaneous Pell Equations

Abstract

Let m, n be integers with $$1<m<n$$ , and let $$d=\gcd (m,n)$$ . In this paper, using the diophantine approximation method, we prove that if $$d \ge 2(n/d)^4$$ , then the simultaneous Pell equations $$x^2-(m^2-1)y^2=1$$ and $$z^2-(n^2-1)y^2=1$$ have only the positive integer solution $$(x,y,z)=(m,1,n)$$ .