Pell方程x^(2)-11y^(2)=1和y^(2)-Dz^(2)=9的公解  

The simultaneous solutions of the Pell equations x^(2)-11y^(2)=1 and y^(2)-Dz^(2)=9

作　　者：牟全武[1] 李立 MU Quanwu;LI Li(School of Science,Xi’an Polytechnic University,Xi’an 710048,China)

机构地区：[1]西安工程大学理学院,陕西西安710048

出　　处：《纺织高校基础科学学报》2023年第3期98-102,共5页Basic Sciences Journal of Textile Universities

基　　金：国家留学基金(202008615008);陕西省自然科学基础研究计划项目(2019JM-337)。

摘　　要：研究了Pell方程x^(2)-11y^(2)=1和y^(2)-Dz^(2)=9的公解问题,这里D为不含平方因子的偶数且至多包含4个奇素因子。利用递推序列、Pell方程及四次丢番图方程解的性质证明了方程当D=2×199时仅有公共正整数解x=3 970,y=1 197,z=60;当D取值为其他情形时仅有公解x=±10,y=±3,z=0。The solutions of the simultaneous Pell equations x^(2)-11y^(2)=1 andy^(2)-Dz^(2)=9 were studied,where D was a square-free even number with at most 4 odd prime factors.By using the recurrent sequence and some properties due to the solutions of Pell's equation and quartic Diophantine equation,it is proved that the equations in title have only the common positive integer solution x=3 970,y=1 197,z=60 in case D=2×199;the equations only have the common integer solutions x=±10,y=±3,z=0 in other cases.

关 键 词：不定方程组 递推序列 正整数解 PELL方程 

分 类 号：O156.7[理学—数学]