椭圆曲线y^2=x(x-p)(x-q)的整数点(Ⅱ)

Integral points on the elliptic curve y^2=x(x-p)(x-q)(Ⅱ)

设p,q为奇素数,m>1为正奇数,且q-p=2~m,q≡11(mod16).证明:当m=3时,椭圆曲线y^2=x(x-p)(x-q)(x>q)无整数点(x,y);当m≥5时,至多有1对整数点(x,y).给出了(p,q)=(11,139)时,椭圆曲线的全部整数点.

Let p ,q be odd primes and m 〉 1 be positive odd number with q - p = 2^m , q 11（mod16） . In this paper, the author proved that ifm=3, then the elliptic curve y^2 =x（x- p）（x -q） （ x 〉 q ） has no integral point （x,y） ; ifm ≥ 5 , then the elliptic curve has at most one integral point （x ,y） . In addition, all integral points of the elliptic curve were given when （p,q） = （11,139） .