关于Pell方程组x^2-42y^2=1与y^2-Dz^2=4的公解

On the Common Solutions of Pell Equations x^2-42y^2=1 and y^2-Dz^2=4

利用同余、递归序列、分解因子、奇偶分析等方法以及解的性质,研究了当D=2p 1…ps(1≤s≤4),其中p 1,…,ps是互异的奇素数时,Pell方程组x^2-42y^2=1与y^2-Dz^2=4的公解。得到除了D=2×337外,该方程组仅有平凡解(x,y,z)=(±13,±2,0)。

By using the methods of congruence,recurrent sequence,decomposition factor,odd and even analysis and the properties of solutions,the solution of Pell equations x^2-42y^2=1 and y^2-Dz^2=4 is studied when D=2p 1,…,p s(1≤s≤4),and p 1,…,p s are diverse odd primes,then only a trivial solution(x,y,z)=(±13,±2,0)is obtained with the exceptions that D=2×337.