关于椭圆曲线y^2=nx(x^2-4)的整数点的一个注记

A Note on the Integral Points on the Elliptic Curve y^2= nx(x^2-4)

运用四次Diophantine方程的性质以及初等方法证明了:设p是素数,当p■1(mod 8)时,方程y^2=px(x^2-4)仅有正整数解(p,x,y)=(3,4,12),(7,16,168),(3,98,1680)(3,6,24),(11,198,9240).若p≡1(mod 8)时,方程y^2=px(x^2-4)至多有一组正整数解.指出了万飞文章中的错误,并利用初等方法巧妙得出了一些新的结论,改进了Wenguan Wu,Alain Togbe,Bo He,Shichun Yang等的解的个数的上界.

In this paper,the properties of quartic Diophantine equations and elementary method are used to prove： let p be a prime,when p 1（mod 8）,（p,x,y） =（3,4,12）,（7,16,168）,（3,98,1680）（3,6,24）,（11,198,9240） are the only positive integer solutions of equation y2= px（x2-4）.If p≡1（mod 8）,the equation y2= px（x2-4） has no more than a set of positive integer solutions. The essay points out the mistakes in the documentary by Wan Fei. And meanwhile,using primary methods,the author skillfully drew some new conclusions and imprve the upper bounds of the number of solutions in documentaries by Wenguan Wu,Alain Togb＇e,Bo He,Shichun Yang.