Pell方程组x^(2)-40y^(2)=1与y^(2)-Dz^(2)=9的公解  

Common solution of Pell equations x^(2)-40y^(2)=1 and y^(2)-Dz^(2)=9

作　　者：贺艳峰[1] 韩帆 李勰 HE Yanfeng;HAN Fan;LI Xie(College of Mathematics and Computer Science,Yan’an University,Yan’an 716000,China)

机构地区：[1]延安大学数学与计算机科学学院,陕西延安716000

出　　处：《延安大学学报（自然科学版）》2024年第2期56-60,共5页Journal of Yan'an University：Natural Science Edition

基　　金：国家自然科学基金项目(12261090,11961072)。

摘　　要：设D=2p_(1)⋯p_(s)(1≤s≤4),其中p_(1),⋯,p_(s)是互不相同的奇素数。主要利用奇偶分析、同余、递归序列以及Pell方程解的性质等初等方法,对Pell方程组x^(2)-40y^(2)=1与y^(2)-Dz^(2)=9的公解进行研究。得出当D≠2×7×103时,该方程组仅有平凡解(x,y,z)=(±19,±3,0);当D=2×7×103时,除了平凡解(x,y,z)=(±19,±3,0)外,还有非平凡解(x,y,z)=(±27379,±4329,±114)。研究结果丰富了这类Pell方程组整数解的研究内容。Let D=2p_(1)⋯p_(s)(1≤s≤4),where p_(1),⋯,p_(s) are distinct odd prime numbers.The common solutions of Pell equations x^(2)-40y^(2)=1 and y^(2)-Dz^(2)=9 are studied by using parity analysis,congruence,recursive sequence and properties of the solutions of Pell equations.It is concluded that when D≠2×7×103,the system have only trivial solution(x,y,z)=(±19,±3,0);when D=2×7×103,in addition to the trivial solution(x,y,z)=(±19,±3,0),there is non-trivial solution(x,y,z)=(±27379,±4329,±114).The results enrich the research content of the integer solution of this kind of Pell equations.

关 键 词：PELL方程 奇偶分析 奇素数 同余 

分 类 号：O156.1[理学—数学]