椭圆曲线整数点方程y^(2)=x^(3)+(m-36)x-6m的解数  

Number of Solutions to the Elliptic Curve Integer Point Equation y^(2)=x^(3)+(m-36)x-6m

作　　者：马莹锐 杨海[1] 曹璐 MA Yingrui;YANG Hai;CAO Lu(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,西安710048

出　　处：《内蒙古农业大学学报（自然科学版）》2024年第2期86-92,共7页Journal of Inner Mongolia Agricultural University(Natural Science Edition)

基　　金：国家自然科学基金项目(11226038,11371012);陕西省自然科学基金项目(2021JM443);陕西省教育厅计划项目(17JK0323)。

摘　　要：设p、q为奇素数,m为正整数,满足p≡5,29(mod 48),q≡1(mod 8),m=12p-72=3q+9,利用Pell方程及四次Diophantine方程的相关结果讨论了椭圆曲线整数点方程y^(2)=x^(3)+(m-36)x-6m的解数问题。证明了该椭圆曲线整数点方程除平凡整数解(x,y)=(6,0),至多有两组非平凡整数解(x,y)=(48ps^(2)t^(2)+6,±24pst(s^(4)+12qt^(4))),其中正整数s、t满足方程(s^(2)-18t^(2))2-48pt^(4)=1。Let p,q be primes andmbe a positive integer with p=5,29(mod48),q=1(mod8)and m=12p-72=3q+9.By us-ing of some correlation results of Pell equation and quartic Diophantine equation,the integral solutions of the elliptic curve integer point equation y^(2)=x^(3)+(m-36)x-6m was studied.It is proved that the elliptic curve integer point equation has at most two inte-gral points(x,y)=(48ps^(2)t^(2)+6,±24pst(s^(4)+12qt^(4))),except for the integral point(x.y)=(6,0),where s and t are positive integers satisfying the equation(s^(2)-18t^(2))2-48pt^(4)=1.

关 键 词：椭圆曲线 DIOPHANTINE方程 LEGENDRE符号 同余 正整数解 

分 类 号：O156.7[理学—数学]