Fibonacci along even powers is (almost) realizable

2020/11/26 by Patrick Moss, Moss, Patrick, Ward, Tom · 2 citations
Computer Science · Materials Science · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #Quasicrystal Structures and Properties #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2011.13068

openalex publication_date 2020/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An integer sequence is called realizable if it is the count of periodic points of some map. The Fibonacci sequence (F_n) does not have this property, and the Fibonacci sequence sampled along the squares (F_n²) also does not have this property. We prove that this is an arithmetic phenomenon related to the discriminant of the Fibonacci sequence, by showing that the sequence (5F_n²) is realizable. More generally, we show that (F_n^2k-1) is not realizable in a particularly strong sense while (5F_n^2k) is realizable, for any k≥1.

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