关于Pell方程组x^2-2y^2=Y^2-bz^2=1的解数

ON THE NUMBER OF SOLUTIONS OF SIMULTANEOUS PELL EQUATIONS x^2-2y^2=y^2-bz^2=1

设a,b是给定且不相等的正整数.我们研究了联立Pell方程组x^2-ay^2=1,y^2-bz^2=1的正整数解(x,y,z)的个数.本文运用Bennett关于联立Padé逼近的一个结果和对数线性型的下界估计,证明了当a=2时,该方程组至多有1组正整数解(x,y,z).

Let a and b be positive integers. In this paper, we study the number of positive integers solutions （x, y, z） of the simultaneous Diophantine equations x^2-ay^2=1,y^2-bz^2=1.It is proved that if a = 2, the above equations possesses at most one positive integer solution （x, y, z） . This result follows from a combination of the techniques including simultaneous Padé approximation to binomial functions, the theory of linear forms in three logarithms of algebraic numbers and computational Diophantine approximations.