Zn上的k次不可约多项式与k阶Carmichael数

设n是合数,如果对一切f（x）∈Zn[x]都满足f（x）nk≡f（x）mod（n,r（x））,那么就称n是模r（x）的k阶Carmichael数,这里r（x）是Zn[x]上的k次首一不可约多项式,用Ck,r（x）表示所有这种数的集合,并且定义Ck=Ur（x）Ck,r（x）.k阶Carmichael数,当k=4时,已证明了n=pq,p,q是不同的奇素数,p2-1,q3-1均整除n4-1,则n∈C4.主要目的是将k=4时得出的结论推广到k≥4的一般情形,利用孙子定理,通过构造Zn上的首一k次不可约多项式f（x）的方法,得出：在k≥4时,设n=pq,如果k=2m,m≥2,pm-1,q2m-1-1均整除nk-1,则n∈Ck;如果k=2m＋1,m≥2,pm-1,pm＋1-1,q2m-1均整除nk-1,则n∈Ck.

Irreducible Polynomials of Order k and Carmichael Numbers of Order k on Zn

Let n be a positive integer and Zn the ring of residues modulo n.Suppose r（x）∈Zn[x] is a monic irreducible polynomial of degree k.We call n a Carmichael number of order k modulo r（x）,if n is composite and f（x）nk≡f（x）mod（n,r（x）） for all f（x）∈Zn[x].Denote the set of all such numbers by Ck,r（x）.Define Ck=Ur（x）Ck,r（x）,where r（x） passes through all monic irreducible polynomials of degree k over Zn.For Carmichael number set of order k,it has been proved that for k=4,if n=pq,where p,q are different odd prime numbers,and p2-1n4-1,q3-1n4-1,then n∈C4.The main purpose of this article is to extend the results which exist when k=4 to more extensive conditions of k≥4.We obtain the following theorem by using the method of the Remainder Theorem and constructing a monic irreducible polynomial of degree k on Zn： for k=4,suppose n=pq,if k=2m,m≥2,pm-1nk-1,q2m-1-1nk-1,then n∈Ck;if k=2m＋1,m≥2,pm-1nk-1,pm＋1-1nk-1,q2m-1nk-1,then n∈Ck.