Periodicity in sequences defined by linear recurrence relations

Abstract

A sequence of rational integers u0, u1, u2, ...(1) is defined in terms of an initial set u0, u1, ..., uk-1 by the recurrence relation un+k + a1un+k-1 + ... + akun = a, n ≥ 0, (2) where a1, a2, ..., ak are given rational integers. The author examines (1) for periodicity with respect to a rational integral modulus m. Carmichael (1) has shown that (1) is periodic for (ak, p) = 1 and has given periods (mod m) for the case where the prime divisors of m are greater than k. The present note gives a period for (1) (mod m) without restriction on m. The results include those of Carmichael. The author also shows that if p divides ak (1) is periodic after a determined number of initial terms and obtains a period.