关于Mordell方程y^2=x^3+k

On the Mordell Equation y^2=x^3+k

设ll,1l,2,…l,s为任意整数,n为正整数,n1,n2,…,ns为任意非负整数.用初等数论方法证明了:如果k满足k=(4l+2)3-∏si=1(4li+1)2ni或k=(4l+3)3-22n∏si=1(4li+1)2ni,则Mordell方程y2=x3+k无整数解.

Let l,l1 ,l2 ,... ,ls be some integer, and n be a positive integer,and n1 ,n2 ,…… ,ns be some nonnegative integer. In this paper, using some elementary number theory methods, we prove that if k satisfies either k=（4l＋2）^3-Пi=1^s（4li＋1）^2ni或k=（4l＋3）^3-2^2nПi=1^s（4li＋1）^2nithen the Mordell y^2=x^3＋k equation has no integral solution.