不定方程x^(2)-72y^(2)=1与y^(2)-Dz^(2)=4的公解

On Common Solutions of Diophantine Equations x^(2)-72y^(2)=1 and y^(2)-Dz^(2)=4

设p_(1),…,p_(s)是不同的奇素数,证明了:当D=2p_(1)…p_(s)(1≤s≤4)时除开D为D=2×577外,不定方程组x^(2)-72y^(2)=1与y^(2)-Dz^(2)=4仅有平凡解(x,y,z)=(±17,±2,0)。

Let p_(1),…,p_(s) are diverse odd primes.In this paper,we prove that if D=2p_(1)…p_(s)(1≤s≤4),then the x^(2)-72y^(2)=1 equations and y^(2)-Dz^(2)=4 have only trivial solutions(x,y,z)=(±17,±2,0),with the exception that D=2×577.