Arithmetic properties of generalized Fibonacci sequences

TL;DR

This paper investigates the arithmetic properties of generalized Fibonacci sequences, resolving conjectures about their valuations, ranks, and distributions, and providing new interpretations and directions for future research.

Contribution

It resolves conjectures on valuations and ranks, offers a new interpretation of the rank modulo prime, and explores the distribution of ranks for specific parameters.

Findings

Resolved conjectures on p-adic valuation and zeta function behavior.

Connected the rank modulo prime to the order in a finite field.

Analyzed the distribution of the rank for t = -1 across different s.

Abstract

The generalized Fibonacci sequences are sequences $\{f_n\}$ which satisfy the recurrence $f_n(s, t) = sf_{n - 1}(s, t) + tf_{n - 2}(s, t)$ ($s, t \in \mathbb{Z}$) with initial conditions $f_0(s, t) = 0$ and $f_1(s, t) = 1$. 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.