虚二次域Q((1-4k^n)^(1/2))类数整除性的例外情况

The Exceptional Cases of the Divisibility of Class Number of the Imaginary Quadratic Field Q((1-4k^n)^(1/2))

设d是无平方因子正整数,h_K是虚二次域K=Q((-d)^(1/2))的类数.又设d满足1+da^2=4k^n,其中a,k,n是适合k>1,n>2的正整数.运用初等数论方法给出了数组(d,a,k,n)可使n|h_K成立的必要条件.

Let d be a positive integer with square free,and let hK denote the class number of imaginary quadratic field K = Q（√-d）.Further let d satisfy 1 ＋ da^2 = 4k^n,where a,k,n are positive integers with k〉 1 and n 2.Using some elemential number theory methods,a necessary condition for（d,a,k,n） satisfies n＋hK is given.