Wen Yu ZHU(1), Qi SUN(), Xian
In this note, we suppose n is a composite, Z_n is a residue class ring mod n, r(x)∈Z_n[x] and r(x) is a monic irreducible polynomial of degree k (k>0) over Z_n. We give a definition for n is Generalized Carmichael Number of order k modulo r(x) and denote this by n∈C_(k,r(x)). So we give another definition: C_k={UC_(k,r(x))|r(x) are all monic irreducible polynomials of degree k (k>0) over Z_n}. Clearly, C_1 is the ordinary Carmichael number. We obtain a necessary and sufficient condition for n∈C_(k,r(x)). Moreover, we get hold of a necessary and sufficient condition for n∈C_k and an easily calculate necessary and sufficient condition for n∈C_2. In addition, we prove C_1(?)C_2 and |C_2|=∞.