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

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) under bar = (a(t))(t >= 0) and (b) under bar = (b(t))(t >= 0) generated by a primitive polynomial of degree n over Z/(M), (a) under bar = (b) under bar if and only if a(t) equivalent to b(t) mod H for all t >= 0, where H > 2 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 n > 6. (C) 2012 Elsevier BM. All rights reserved.