椭圆曲线y^2=x(x-p)(x-q)的整数点(Ⅰ)

Integral Points on the Elliptic Curve y^2=x(x-p)(x-q)(Ⅰ)

设P，q为奇素数，m为正奇数，且P＋2^m=q，P=3（mod4）．证明了：当m=1或3时，椭圆曲线y^2=x（x—p）（z—q）（z〉q）至多有1对整数点（X，可）；当m≥5时，该椭圆曲线至多有2对整数点（x，y）．同时具体给出了（p，q）=（71，103）时椭圆曲线的全部整数点．

Let p,q be odd primes and m be positive odd number with p ＋2m = q,p = 3（rood4）. In this paper, we prove that if m= 1 or 3, then the elliptic curve y^2 = x（x - p）（x - q）（x 〉 q） has at most one integral point （x,y）; if m ≥5 , then the elliptic curve has at most two integral points （x, y）. Additionary, all integral points of the elliptic curve when （p, q） = （71,103） are given.