椭圆曲线y^2=x^3-px的整数点

The Integral Points on Elliptic Curves y^2=x^3-px

设p是奇素数,N(p)是椭圆曲线y2=x3-px上的非平凡整数点(x,±y)的个数.根据二次和四次Diophantine方程的性质证明了:当p≡1(mod 8)时,N(p)≤3;当p≡5(mod 8)时,N(p)≤2;当p≡7(mod 8)时,N(p)≤1;当p≡3(mod 8)时,N(p)=0.

Let pbe an odd prime,and let N（p）denote the number of non-trivial integer points（x,±y）on the elliptic curve y2=x3-px.In this paper,using some properties of quadratic and quartic diophantine equations,we prove that if p≡1（mod 8）,then N（p）≤3;if p≡5（mod 8）,then N（p）≤2;if p≡7（mod 8）,then N（p）≤1;if p≡3（mod 8）,then N（p）=0.