On Sequences Defined by Linear

where a, ai, a2> ■ ■ ■ , a* 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. Carmichaelî 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. Definition 1. We say that w is a general period of the recurrence (2) for the modulus m if every sequence of rational integers satisfying (2) has the period -rr (mod m). Theorem 1. The minimum period p (mod m) of a sequence (1) satisfying (2) is a divisor of any general period it (mod m) of (2). For, since (1) has the period w, ir^p. Suppose p does not divide ir, that is, 7T = c7p+p, where 0<p<p. Then w¿+í(1+<)=w¿ (mod m), that is, wi+p=w¿ (mod m) and (1) has the period p, which is contradictory. The algebraic equation