Irreducible Polynomials of Order k and Carmichael Numbers of Order k on Z_n
Qin Shi-xia · Journal of Chengdu University of Information Technology · 2010
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.