关于不定方程组x^2-12y^2=1与y^2-Dz^2=4的公解

On the System Diophantine Equations x^2-12y^2=1 and y^2-Dz^2= 4

利用递归序列、Pell方程的解的性质,证明了D=2~n(n∈Z^+)时,不定方程x^2-12y^2=1与y^2-Dz^2=4只有平凡解(x,y,z)=(±7,±2,0)。

By using recursive sequence and some properties of the solution to Pell equation,the following were proved： If D=2^n（ n∈Z^＋）,the system Diophantine equations x^2-12y^2= 1 and y^2-Dz^2= 4 has only trivial solution（ x,y,z） =（ ± 7,± 2,0）.