On sequences defined by linear recurrence relations

Abstract

where a, a,, a2, * * , ak are given rational integers. The purpose of this paper is to investigate the periodicity of such sequences with respect to a rational integral modulus m. Carmichaelt has studied the period for a modulus m whose prime divisors exceed k and are prime to ak. In this paper, I give a solution to the problem without restriction on m. If m is prime to ak the sequence (1) is periodic from the start; otherwise, it is periodic after a definite number of initial terms.