On Carmichael Numbers Of Order 3

Let r(x)∈Z[x]be a monic irreducible polynomial of degreeκ(≥1)and let n be a positive composite integer.If r(x)is also irreducible in Z_n[x]and f(x)^nκ≡f(x)mod(n,r(x))for all f(x)∈Z_n[x]then we call n a Carmichael number of orderκmodulo r(x).Denote the set of all such numbers by C_κ,r(x).Define C_κ=∪_r(x)C_κ,r(x),where r(x)passes through all monic irreducible polynomials over Z of degreeκ.We call elements of the set C_κCarmichael numbers of orderκ.ZHU and SUN[Journal of Sichuan University(Natural Science Edition),42;1(2005), 47-51]first gave a necessary condition for Carmichael numbers of order 3.Then they gave a sufficient condition for Carmichael numbers of order 3,but did not find any such numbers less than 10⁸ satisfying the sufficient condition.At last,they asked three questions;1.is the necessary condition also sufficient? 2.are there any such numbers＞10⁸ satisfying their sufficient condition? 3.is it true that #C₃=∞?In Chapter 3 of this thesis,we answer Question 1 affirmatively.We first prove that the necessary condition is also sufficient and generalize it to a suffi-cient condition for Carmichael numbers of orderκ.Using this equivalent condition for Carmichael numbers of order 3,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 10⁸ 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 the above sufficient condition.In Chapter 4,we try to answer Questions 2 and 3.We first remark that answers to the two questions should be affirmative,or more precisely,we remark that there should be infinitely many Carmichael numbers of order 3 satisfying certain condi-tions and that there should even be infinitely many Carmichael numbers of order 3, by Howe's heuristic argument[Mathematics Of Computation,69;232(2000),1711-1719]for rigid Carmichael numbers of orderκ.Then we actually answer Question 2 affirmatively by presenting a method for finding examples of Carmichael numbers of order 3 satisfying certain conditions and using the method to provide hundreds of such examples.Our method is an analog to Howe's one for finding rigid Carmichael numbers of order 2.