Periodicity of power Fibonacci sequences modulus a Fibonacci number

2022/04/01 by Josep M. Brunat, Brunat, Josep M., Joan-C. Lario +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #2020: 11B39 11B50 11A15 #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Fractal and DNA sequence analysis #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2204.00234

openalex publication_date 2022/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathcal F=(F_i:i≥ 0) be the sequence of Fibonacci numbers, and j and e be non negative integers. We study the periodicity of the power Fibonacci sequences \mathcal F^e(F_j)=(F_i^e\pmodF_j: i≥ 0). It is shown that for every j,e≥ 1 the sequence \mathcal F^e(F_j) is periodic and its periodicity is computed. The result was previously known for \mathcal F(F_j); that is, for e=1. For e∈ \1, 2\, the values of the normalized residues ρ_i≡ F_i^e\pmodF_j with 0≤ ρ_i

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