指数不定方程2^(x)+(pq)^(y)=z^(2)的非负整数解  

Non-negative integer solution of the exponential Diophantine equation 2^(x)+(pq)^(y)=z^(2)

作　　者：蒋玉婷 管训贵 JIANG Yuting;GUAN Xungui(School of Mathematics and Physics,Taizhou University,Taizhou 225300,China)

机构地区：[1]泰州学院数理学院,江苏泰州225300

出　　处：《高师理科学刊》2024年第1期8-11,共4页Journal of Science of Teachers＇College and University

基　　金：国家自然科学基金项目(11471144);江苏省教育科学“十三五”规划课题(D20200115);江苏省“青蓝工程”数学教育教学团队资助项目(SJS201903)。

摘　　要：设p,q为奇素数,p<q.利用同余式和平方剩余的方法,对不定方程2^(x)+(pq)^(y)=z^(2)的非负整数解进行了研究.证明得出:若(p,q)≡(1,3),(1,5),(3,1),(3,5),(3,7),(5,1),(5,3),(5,7),(7,3),(7,5)(mod8),则不定方程2^(x)+(pq)^(y)=z^(2)除p=3,q=5时仅有非负整数解(x,y,z)=(3,0,3),(0,1,4),(6,2,17)以及q=p+2,p≠3时仅有非负整数解(x,y,z)=(3,0,3),(0,1,p+1)外,都仅有非负整数解(x,y,z)=(3,0,3).Let p,q are odd primes and p<q.The non-negative integer solution of the Diophantine equation 2^(x)-(pq)^(y)=z^(2) was researched with the primary methods of congruence and quadratic remainder.It was proven that if(p,q)º(1,3),(1,5),(3,1),(3,5),(3,7),(5,1),(5,3),(5,7),(7,3),(7,5)(mod8),then the Diophantine equation 2^(x)-(pq)^(y)=z^(2) has only non-negative integer solution(x,y,z)=(3,0,3),with the exceptions that the equation has only non-negative integer solutions(x,y,z)=(3,0,3),(0,1,4),(6,2,17)when p=3,q=5 and the equation has only non-negative integer solutions(x,y,z)=(3,0,3),(0,1,p+1)when q=p+2,p≠3.

关 键 词：指数不定方程 非负整数解 Catalan猜想 同余 平方剩余 

分 类 号：O156.4[理学—数学]