Pell方程组x^(2)-(k^(2)+k)y^(2)=1,y^(2)-bz^(2)=4的公解

Complete Solutions of the Simultaneous Pell's Equations x^(2)-(k^(2)+k)y^(2)=1,y^(2)-bz^(2)=4

利用Pell方程解的性质、递归序列和一些四次方程的结果,研究了k=7时,Pell方程组x^(2)-(k^(2)+k)y^(2)=1,y^(2)-bz^(2)=4(b=2p 1…p s(1≤s≤4),(p 1,…,p s,是互异的奇素数)的公解,得到除了b=2×449外,该方程组仅有平凡解(x,y,z)=(±15,±2,0).

Let k=7,we get that the equation system x^(2)-(k^(2)+k)y^(2)=1,y^(2)-bz^(2)=4(b=2p 1…p s(1≤ s≤4),(p 1,…,p s,are mutually different odd prime numbers)has only trivial solutions except b=2×449 by using only the properties of the Pell equation solution,recursive sequences and some quartic equations.