椭圆曲线y2=2px(x2+1)上正整数点的个数

设p是奇素数,N(p)是椭圆曲线E:y2=2px(x2+1)的正整数点(x,y)的个数.主要讨论了N(p)的性质,运用初等方法及四次Diophantine方程的性质,对某些特殊素数p,给出了N(p)的上界.证明了当p≡1(mod 8)且p=s2+32t,其中s,t是正整数时,N(p)≤3;当p≡1(mod 8)且p+s...

The number of positive integral points on the elliptic curve y~2 = 2px(x~2 + 1)

Let p be an odd prime,and let N(p) denote the number of positive integral points(x,y) on the elliptic curve E: y2 = 2px(x2 +1).The main purpose of this paper is to study the properties of N(p).Using the elementary method and the properties of the quartic diophantine equations.A new upper bound estimate for N(p) are given.Proved that if p ≡ 1(mod 8) and p = s2 +32t2,where s,t are positive integers,then N(p) ≤ 3;If p ≡ 1(mod 8) and p = s2 + 32t2,then N(p) ≤ 2;If p ≡ 5 or 7(mod 8),then N(p) ≤ 1;If p ≡ 3(mod 8),then N(p) = 0.These works are materialization of early results.