关于同余式sum from i=1 to k(x_i^2)≡o(mod p)解的个数与二次域Q(p^(1/2))的类数

ON THE NUMBER OF SOLUTIONS OF THE CONGRUENCE sum from i=1 to k(x_i^2)≡o(MOD p) AND THE CLASS NUMBER OF Q(p^(1/2))

最近,Takashi Agoh对于素数p≡1(mod 4)给出了计算二次域Q(p^(1/2))的类数h的一个公式,此公式仅依赖Q(p^(1/2))的基本单位∈,素数p以及数α=1+(?)(-1)N_k,其中N_k为同余式x_1~2+…+x_k^2≡0(mod p),1≤x_1<x_2<…<x_k≤p-1/2的解(x_1,…,X_k)的个数.本文对奇数p(不限于p≡1(mod 4)),得到N_k的若干性质和计算N_2,N_3的公式.

Let p be an odd prime with p=1 (mod 4) and the fundamental unit of real quadratic field over the rationals , Also let h be the class mumber of Recently, Takashi Agoh prove thatlogwhere Let denote the number of solutions (X1,xk) of congruencex12+ xk2 =0 (mod p),In this paper, we obtain some properties of Np and formulae computing N2, Ns, where p is a odd prime