【摘要】 设Ck(k>0)表示k阶Carmichael数集,C1即为通常的Carmichael数集.作者考虑3阶Carmichael数的性质,得到了n∈C3的一个必要条件(定理1)和两个容易计算的充分条件(定理2和定理3).对于108以下,发现了43个3阶Carmichael数.同时,验证了在108以下不存在满足定理2中条件的3阶Carmichael数,以及在104以下仅有3个3阶Carmichael数:1885,2101,9529.还证明了C1 C3以及C2 C3,部分回答了RajatBhattacharjee等提出的一个问题.对于3阶Carmichael数,作者提出了三个未解决的问题.更多还原

【Abstract】 Let C_k（k>0） denote the set of Carmichael numbers of order k and C₁ be the set of the standard Carmichael numbers. In this paper, the authors consider C₃ and get a necessary condition for Carmichael numbers of order 3. Two easily computable sufficient conditions for Carmichael numbers of order 3 are also given. The authors find 43 Carmichael numbers of order 3 less than 10⁸ and it is verified that one type of Carmichael numbers of order 3 less than 10⁸ does not exist （Theorem 2）, and they prove that there are only three Carmichael numbers of order 3 less than 10⁴, which are 1885, 2101, 9529. Furthermore, they prove that C₁C₃ and C₂C₃, which gives a partial answer to an unknown problems.更多还原