Divisibility Properties of Recurrent Sequences

Abstract

Let m, r ∈ N. We will show, that the recurrent sequences xn = xnrn−1 + 1 (mod g),  xn = xn!n−1 + 1 (mod g)  and xn = xrnn−1 + 1 (mod g) are periodic modulo m, where m ∈ N, and we will find some estimations of periods and pre-periodic parts. Later we will give an algorithm sophisticated enough for finding periods length in polynomial time.