椭圆曲线y^2=7nx(x^2-8)的整数点

Integer Points of Elliptic Curves y^2=7nx(x^2-8)

利用同余的性质、奇偶数的性质、Legendre符号的性质等初等方法,证明了n≡±5(mod8)为奇素数时椭圆曲线y^2=7nx(x^2-8)除整数点(x,y)=(0,0)外至多有4个整数点。研究结果对于a,b∈Z时,椭圆曲线y^2=pqx(x^2-a)的求解有一定的借鉴作用,也推进了该类椭圆曲线的研究。

The integral points on elliptic curve is a very important problem of Number Theory and related field.The positive integral points on elliptic curve y^2=7nx(x^2-8) remains unresolved.In this paper,the elementary methods were used including some properties of congruence,some properties of odd number and even number,and Legendre symbol.It was proved that the elliptic curve in title at most have four integer points except(x,y)=(0,0).These results can help solve the integral points on elliptic curve y^2=(x+a)(x^2+ax+b),a,b∈Z,and these results promote the study of the kind of elliptic curve.