$p$-regularity of the $p$-adic valuation of the Fibonacci sequence

TL;DR

This paper proves that the p-adic valuation of Fibonacci numbers forms a p-regular sequence for all primes p and determines its rank for most primes, linking it to Fibonacci periodicity modulo m.

Contribution

It establishes the p-regularity of the p-adic valuation of Fibonacci numbers and explicitly calculates the sequence's rank for primes other than 2 and 5.

Findings

p-adic valuation sequence is p-regular for all primes p

The rank of the sequence is α(p) + 1 for p ≠ 2, 5

Connects sequence rank to Fibonacci periodicity modulo m

Abstract

We show that the $p$-adic valuation of the sequence of Fibonacci numbers is a $p$-regular sequence for every prime $p$. For $p \neq 2, 5$, we determine that the rank of this sequence is $\alpha(p) + 1$, where $\alpha(m)$ is the restricted period length of the Fibonacci sequence modulo $m$.