# Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients II

## TL;DR

This paper derives explicit formulas for the p-adic valuations of Fibonomial coefficients for all primes and positive integers, extending previous results and characterizing divisibility conditions for these coefficients.

## Contribution

It provides comprehensive formulas for p-adic valuations of Fibonomial coefficients for all primes and extends prior work limited to specific primes and cases.

## Key findings

- Explicit formulas for p-adic valuations of Fibonomial coefficients.
- Characterization of divisibility of Fibonomial coefficients by primes.
- Extension of previous results to all primes and positive integers.

## Abstract

In this article, we give explicit formulas for the $p$-adic valuations of the Fibonomial coefficients ${p^a n \choose n}_F$ for all primes $p$ and positive integers $a$ and $n$. This is a continuation from our previous article extending some results in the literature, which deal only with $p = 2,3,5,7$ and $a = 1$. Then we use these formulas to characterize the positive integers $n$ such that ${pn \choose n}_F$ is divisible by $p$, where $p$ is any prime which is congruent to $\pm 2 \pmod{5}$.