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

Abstract

Let be a square-free odd integer and the integer residue ring modulo . This paper studies the distinctness of primitive sequences over modulo 2. Recently, for the case of , a product of two distinct prime numbers and , the problem has been almost completely solved. As for the case that 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 over is distinct modulo 2, where is a positive integer. Besides as an independent interest, this paper also involves two distribution properties of primitive sequences over , which are related closely to our main results.

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

“…Based on some estimates of exponential sums over integer residue rings, a sufficient condition for Assumption 1 to be valid for (M, n) was given in our previous paper [19,Theorem 11]. Experimental data shows that the condition is satisfied for most M's if n > 6, see [19, p. 15, Table 1] for more details.…”

“…Recently research interests on primitive sequences over Z/(p e ) are further extended to primitive sequences over Z/(M) [1,18,19], where M is a square-free odd integer. One of important reasons for this is that the period of a primitive sequence a of order n over Z/(p e ) is undesirable if e 2.…”

“…For the case where M is a product of more odd prime numbers, the distinctness of primitive sequences over Z/(M) modulo 2 has been quite resistant to proof. Until now only a class of primitive sequences of order 2k + 1 (k 1) over Z/(M) are proved to be distinct modulo 2 in [19].…”

“…Let H > 2 be an integer divisible by a prime number coprime with M. In this paper, under an assumption that every element in Z/(M) occurs in a primitive sequence of order n over Z/(M), it is proved that modulo H reductions of primitive sequences generated by a primitive polynomial of degree n over Z/(M) are pairwise distinct. The assumption is known to be valid for most M's if n > 6, see [19,Theorems 11 and 13] for more details.…”

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

“…The case that N is an odd prime power integer has been completely solved in [3]. Besides, several results for square-free N can be found in [4][5][6][7][8]. But there is no known result in the case when N is neither square-free nor prime power.…”