2014/07/30 by Soo-Hyun Park, Park, Soohyun
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1407.8086
openalex publication_date 2014/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The generalized Fibonacci sequences are sequences \fn\ which satisfy the recurrence fn(s, t) = sfn - 1(s, t) + tfn - 2(s, t) (s, t ∈ ℤ) with initial conditions f0(s, t) = 0 and f1(s, t) = 1. In a recent paper, Amdeberhan, Chen, Moll, and Sagan considered some arithmetic properites of the generalized Fibonacci sequence. Specifically, they considered the behavior of analogues of the p-adic valuation and the Riemann zeta function. In this paper, we resolve some conjectures which they raised relating to these topics. We also consider the rank modulo n in more depth and find an interpretation of the rank in terms of the order of an element in the multiplicative group of a finite field when n is an odd prime. Finally, we study the distribution of the rank over different values of s when t = -1 and suggest directions for further study involving the rank modulo prime powers of generalized Fibonacci sequences.