关于椭圆曲线y^2=px(x^2+2)的整点研究  

Integral Points on Elliptic Curve y^2=px(x^2+2)

作　　者：印婧婧 管训贵 

机构地区：[1]泰州学院数理学院,江苏泰州225300

出　　处：《河北北方学院学报（自然科学版）》2017年第7期11-13,共3页Journal of Hebei North University:Natural Science Edition

基　　金：江苏省教育科学"十二五"规划课题(D201301083);云南省教育厅科研课题(2014Y462);泰州学院教授基金项目(TZXY2015JBJJ002)

摘　　要：目的针对数论和算术代数几何学的有趣问题——椭圆曲线整点的确定,研究椭圆曲线G:y^2=px(x^2+2)的整点。方法运用二次和四次Diophantine方程的性质。结果设s是正整数,则当素数p=8(18s^2-18s+1)(9s^2-9s+1)+3时,椭圆曲线G至多有1个正整数点;当p=32s^4+1时,椭圆曲线G仅有1个正整数点(x,y)=(8s^2,128s5+4s)。结论解决了椭圆曲线G的可解性问题。即对某些特殊的素数P,椭圆曲线G至多有1个正整数点。所获命题,提供了研究椭圆曲线整点问题的一个思路。Objective For an interesting problem in number theory and arithmetic algebraic geometry——the determination of integral points on elliptic curve,the points on the elliptic curve G:y^2=px(x^2+2)are studied.Methods Using properties of quadratic and quartic Diophantine equations.Results Let s be positive integer.If pbe prime with p=8(18s^2-18s+1)(9s^2-9s+1)+3,then the elliptic curve G has at most one positive integral ponit;if p=32s^4+1,then the elliptic curve G has only one positive integral point(x,y)=(8s^2,128s5+4s).Conclusion The study proves the solvability of the elliptic curve G,that is the elliptic curve G has at most one positive integral point for some special prime p.The statements supply an idea to study the problem of integral points on elliptic curve.

关 键 词：椭圆曲线 整数点 解数 上界 

分 类 号：O156.1[理学—数学]