On the Distinctness of Binary Sequences Derived From Primitive Sequences Modulo Square-Free Odd Integers

Let M be a square-free odd integer and Z/(M) the integer residue ring modulo M . This paper studies the distinctness of primitive sequences over Z/(M) modulo 2. Recently, for the case of M=pq, a product of two distinct prime numbers p and q, the problem has been almost completely solved. As for the case that M is a product of more prime numbers, the problem has been quite resistant to proof. In this paper, a partial proof is given by showing that a class of primitive sequences of order 2n^´+1 over Z/(M) is distinct modulo 2, where n^´ is a positive integer. Besides as an independent interest, this paper also involves two distribution properties of primitive sequences over Z/(M), which are related closely to our main results.