关于3阶Carmichael数的注记(英文)  

Notes on Carmichael Numbers of Order 3

作　　者：刘亚[1] 

机构地区：[1]安徽师范大学数学系,芜湖241000

出　　处：《四川大学学报（自然科学版）》2006年第6期1197-1201,共5页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(10071001);安徽省自然科学基金(01046103);安徽省教育厅自然科学基金(2002KJ131)

摘　　要：如果合数n对于所有f(x)∈Zn[x]都有f(x)nk≡f(x)mod(n,r(x))成立,就称n是模r(x)的k阶Carmichael数,这里r(x)∈Zn[x]是k次首一不可约多项式,用Ck,r(x)表示所有的这种数的集合.定义Ck=∪r(x)Ck,r(x),这里r(x)跑遍Zn[x]中所有k次首一不可约多项式.Ck里面的元素就称为k阶Carmichael数.2005年,朱文余和孙琦首先给出了3阶Carmichael数的一个必要条件(1),然后又给出了这种数的一个充分条件(2),并发现108内没有满足条件(2)的这种数.最后他们问必要条件(1)是否也是充分的,还问108以外是否有满足充分条件(2)的这种数?本文作者首先证明了朱和孙给出的必要条件(1)也是充分的,然后利用这个等价条件搜索到所有小于3037000499的3阶Carmichael数,共713个,其中149个小于108(包括朱和孙找到的43个).这713个数均不满足朱和孙给出的充分条件(2).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 k modulo r （x） , if n is composite and f（x）^nk≡f（x） rood （ 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. We call elements of the set Ck Carmichael numbers of order k . In 2005, ZHU and SUN first gave a necessary condition （1） for Carmichael numbers of order 3. Then they gave a sufficient condition （2） for Carmichael numbers of order 3, but did not find any such numbers less than 10^8 satisfying condition （2）. At last, they asked if the necessary condition （1） is also sufficient and asked if there were any such numbers beyond 10^8 satisfying condition （2）. In this paper, we first prove that the necessary condition （1） is also sufficient. Using this equivalent condition （1）, we then describe a procedure for finding all Carmichael numbers of order 3 less than 3037000499. There are in total 713 such numbers, 149 of them are less than 10^8 including 43 ones found by ZHU and SUN. At last we give an overview of the 713 numbers, and tabulate 35 of them, which have six prime factors. There are no Carmichael numbers of order 3 less than 3037000499 satisfying condition （2）.

关 键 词：3阶Carmichael数 模n剩余类环上的不可约多项式 孙子定理 

分 类 号：O156.2[理学—数学]