广义Carmichael数

设n是一个合数,Z_n表示模n的剩余类环,r(x)∈Z_nx]是一个首一的k(>0)次不可约多项式。本文引入n是k阶摸r(x)的Carmichael数的定义,全体这样的数记为集C_(k,r)(x),由此给出k阶Carmichael数集:C_k={∪C_(k,r)(x)|r(x)过全体Z_n上的首一k次不可约多项式}。显然C_1表示通常的Carmichael数集。作者得到了n∈C_(k,r(x))的一个充要条件,进而得到n∈C_k的一个充要条件及n∈C_2的一个更易计算的充要条件,还证明了C_1(?)C_2以及|C_2|=∞。

Generalized Carmichael Numbers

In this note, we suppose n is a composite, Z_n is a residue class ring mod n, r(x)∈Z_nx] and r(x) is a monic irreducible polynomial of degree k (k>0) over Z_n. We give a definition for n is Generalized Carmichael Number of order k modulo r(x) and denote this by n∈C_(k,r(x)). So we give another definition: C_k={UC_(k,r(x))|r(x) are all monic irreducible polynomials of degree k (k>0) over Z_n}. Clearly, C_1 is the ordinary Carmichael number. We obtain a necessary and sufficient condition for n∈C_(k,r(x)). Moreover, we get hold of a necessary and sufficient condition for n∈C_k and an easily calculate necessary and sufficient condition for n∈C_2. In addition, we prove C_1(?)C_2 and |C_2|=∞.