关于Pell方程x^2─2y^2＝1与y^2─Dz^2＝4的公解

Solutions on the Pell equation x￣2-2y￣2=1,y￣2-Dz￣2=4

本文研究Ｐｅｌｌ方程ｘ￣２─２ｙ￣２＝１与ｙ￣２─ＤＺ￣２＝４的公解的问题，完整地证明了当Ｄ无平方因子且至多含三个不同奇素因子时，除开（ｘ，ｙ，ｚ）＝（１７，１２，２）．（Ｄ＝３５）；（ｘ，ｙ，ｚ）＝（１９６０１，１３８６０．２６）．（Ｄ＝２９×４１×２３９）外无其它非平凡解．这个结果加强了Ｍａｈａｎｔｙ￣［１］和陈建华￣［２］的结论．

In this paper,we study the simultaneous diophantine equation x￣2-2y￣2=1andy￣2-Dz￣2=4.we proued the following theorems. Theorem Let D=p_1p_2p_3,p_1p_2orp_1,p_i(i=l,2,3)are dissimilar primes,then the equationx￣2-2y￣2=land y￣2-Dz=4 hes only solutions(x,y,)=(17,12,2),(D=35);(x,y,z)=(19601 ,13860,26),(D=29× 41 × 239).