椭圆曲线y^2=qx(x^2-256)的正整数解  

The positive integer solutions of elliptic curves y^2=qx(x^2-256)

作　　者：谢甜甜 杨海[1] 许倩 XIE Tiantian;YANG Hai;XU Qian(School of Sciences,Xi’an Polytechnic University,710048,Xi’an,Shaanxi,PRC)

机构地区：[1]西安工程大学理学院,陕西省西安市710048

出　　处：《曲阜师范大学学报（自然科学版）》2021年第1期47-51,共5页Journal of Qufu Normal University(Natural Science)

基　　金：国家自然科学基金(11226038,11371012);陕西省自然科学基金(2017JM1025);陕西省教育厅项目(17JK0323).

摘　　要：利用初等数论方法及同余性质证明了椭圆曲线方程y^2=qx(x^2-256)除整数解(0,0),(16,0)外还有其它正整数解,即:(ⅰ)当q=5时方程仅有正整数解(x,y)=(20,120),(144,3840);(ⅱ)当q=29时方程仅有正整解(x,y)=(156816,334414080);(ⅲ)当q=41时方程仅有正整数解(x,y)=(25,615);(ⅳ)当q≠5,29,41时方程至多只有一组正整数解(x,y),其中q为无平方因子的正奇数.By using the properties of congruence and the methods of elementary number theory,it is proved that the elliptic curve y^2=qx(x^2-256)has other positive integer points except integer solutions(0,0)and(16,0),that is to say:(ⅰ)If q=5,then the elliptic curve in title has integer points(x,y)=(20,120),(144,3840).(ⅱ)If q=29,then it has integer point(x,y)=(156816,334414080).(ⅲ)If q=41,then it has integer point(x,y)=(25,615).(ⅳ)If q≠5,29,41,then it has at most one positive integer solution(x,y),where q is a positive odd number without squared factor.

关 键 词：同余 正整数解 椭圆曲线 

分 类 号：O156.7[理学—基础数学]