On the distinctness of primitive sequences over Z/(p(e)q) modulo 2

摘要

Let N be an integer greater than 1 and Z/(N) the integer residue ring modulo N. Extensive experiments seem to imply that primitive sequences of order n >= 2 over Z/(N) are pairwise distinct modulo 2. However, efforts to obtain a formal proof have not been successful except for the case when N is an odd prime power integer. Recent research has mainly focussed on the case of square-free odd integers with several special conditions. In this paper we study the problem over Z/(p(e)q), where p and q are two distinct odd primes, e is an integer greater than 1. We provide a sufficient condition to ensure that primitive sequences generated by a primitive polynomial over Z/(p(e)q) are pairwise distinct modulo 2.