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.
Highlights
IntroductionThe reader may consult [3] for the latest developments in this problem
The study of recurrent sequences (in particular, the sequence given by xn+1 = xfn(n) + 1, where limn→∞ f (n) = ∞) was motivated by the construction of some special transcendental numbers ζ for which the sequences of their integral parts [ζn], n = 1, 2, 3, . . . , have some divisibility properties [1], [2]
The reader may consult [3] for the latest developments in this problem. It was proved in [4] that the sequence given by x1 ∈ N and xn+1 = xnn+1 + P (n) for n ≥ 1, where P (z) is an arbitrary polynomial with integer coefficients, is periodic modulo g for every g ≥ 2
Summary
The reader may consult [3] for the latest developments in this problem It was proved in [4] that the sequence given by x1 ∈ N and xn+1 = xnn+1 + P (n) for n ≥ 1, where P (z) is an arbitrary polynomial with integer coefficients, is periodic modulo g for every g ≥ 2. It was proved in [5] that the sequence given by x1 ∈ N and xn+1 = F Are periodic with periods T1 ≤ gφ(g), T2 ≤ 2, T3 ≤ gφ(φ(g)), respectively, where φ stands for Euler’s totient function We use this information to build the algorithm and find its estimation
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