ON r-ORDER RECURRENCES*

with initial terms u0 ux = ••• = ur_2 = 0, ur_1 1. Then (u) is called a unit sequence with coefficients al5 a25 -.., ar. For a positive integer AT, the primitive period of (u) modulo AT, denoted by K(M), is the least positive integer m such that un + m = un (mod AT) for all nonnegative integers n greater than or equal to some fixed integer n0. It is known that the primitive period modulo M of a unit sequence (u) is a period modulo M of any other recurrence satisfying the same recursion relation (see [4], pp. 603-04). The rank of (u) modulo M, denoted by k(M), is the least integer m such that un + m n (mod AT) for some residue s and for all integers n greater than or equal to some fixed nonnegative integer nQ. We call s the principal multiplier of (u) modulo AT. If (ar, AT) = 1, then it is known from [1] that (u) is purely periodic modulo M and K(M) \k{M) . Furthermore, if (ar, AT) = 1, Carmichael [1] has shown that the principal multiplier s is a unit modulo M and K(M) /k(M) = fi^AT) is the exponent of the multiplier s modulo M. In this paper, we will put constraints on K(M) given k(M) and the exponent of av modulo AT. Our two main results are Theorems 1 and 2. Theorem 2 is a refinement of Theorem 1.