On the distinctness of modular reductions of primitive sequences modulo square-free odd integers

Authors:
Qunxiong Zheng;Wenfeng Qi;Tian Tian
Affiliations:
Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou, China;Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou, China;Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou, China
Venue:
Information Processing Letters
Year:
2012

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Abstract

Let M be a square-free odd integer with at least two different prime factors and Z/(M) the integer residue ring modulo M. In this paper, it is shown that for two primitive sequences a@?=(a(t))"t"="0 and b@?=(b(t))"t"="0 generated by a primitive polynomial of degree n over Z/(M), a@?=b@? if and only if a(t)=b(t)modH for all t=0, where H2 is an integer divisible by a prime number coprime with M. This result is obtained basing on the assumption that every element in Z/(M) occurs in a primitive sequence of order n over Z/(M), which is known to be valid for most M@?s if n6.