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)Z/(M) the integer residue ring modulo M  . In this paper, it is shown that for two primitive sequences a̲=(a(t))t⩾0 and b̲=(b(t))t⩾0 generated by a primitive polynomial of degree n   over Z/(M)Z/(M), a̲=b̲ if and only if a(t)≡b(t)modH for all t⩾0t⩾0, where H>2H>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)Z/(M) occurs in a primitive sequence of order n   over Z/(M)Z/(M), which is known to be valid for most M  ʼs if n>6n>6.

► Primitive sequences over Z/(M)Z/(M) are shown to be distinct modulo H.
► The result is obtained basing on an assumption of primitive sequences of order n   over Z/(M)Z/(M).
► The assumption is known to be valid for most M  ʼs if n>6n>6.