关于不定方程组x^2-14y^2=1与y^2-Dz^2=16的公解  

On the solutions of the simultaneous Diophantine equations x^2-14y^2=1 and y^2-Dz^2=16

作　　者：瞿云云 曾吉文[1] QU Yunyun;ZENG Jiwen(School of Mathematical Sciences,Xiamen University,Xiamen 361005,China;School of Mathematical Sciences,Guizhou Normal University,Guiyang 550001,China)

机构地区：[1]厦门大学数学科学学院,福建厦门361005 [2]贵州师范大学数学科学学院,贵州贵阳550001

出　　处：《厦门大学学报（自然科学版）》2020年第4期512-515,共4页Journal of Xiamen University：Natural Science

基　　金：国家自然科学基金(61562012);贵州省科学技术基金(黔科合基础[2019]1221号)。

摘　　要：设p 1,p 2,…,p s(1≤s≤4)是互异的奇素数,利用递归数列、Pell方程解的性质证明了当D=2p 1p 2…p s(1≤s≤4)时,不定方程组x^2-14y^2=1与y^2-Dz^2=16的整数解如下:当D=2×449时,方程组仅有解(x,y,z)=(±13455,±3596,±120)以及解(x,y,z)=(±15,±4,0);当D≠2×449时,方程组仅有解(x,y,z)=(±15,±4,0).Let p 1,p 2,…,p s(1≤s≤4)be distinct odd primes.By using recurrent sequence and properties of the solution of the Pell equation,we concluded that,if D=2p 1p 2…p s(1≤s≤4),then the Diophantine equations x^2-14y^2=1 and y^2-Dz^2=16 lead to following integer solutions:when D=2×449,the equations are only associated with solutions(x,y,z)=(±13455,±3596,±120)and(x,y,z)=(±15,±4,0),when D≠2×449,the equations are only associated with solutions(x,y,z)=(±15,±4,0).

关 键 词：不定方程组 递归数列 整数解 PELL方程 

分 类 号：O156[理学—数学]