# The 2-adic valuation of generalized Fibonacci sequences with an application to certain Diophantine equations

## TL;DR

This paper investigates the 2-adic valuation of generalized Fibonacci sequences, derives bounds for solutions to related Diophantine equations, and explores connections with p-regular sequences, advancing understanding of these number sequences.

## Contribution

It provides explicit formulas for the 2-adic valuation of 2k-nacci sequences and applies these to solve specific Diophantine equations, extending to a broader family of sequences.

## Key findings

- Explicit 2-adic valuation formulas for 2k-nacci sequences
- Bounds on factorials expressed as products of these sequences
- Potential links between these sequences and p-regular sequences

## Abstract

In this paper we focus on finding all the factorials expressible as a product of a fixed number of $2k$-nacci numbers with $k \geq 2$. We derive the 2-adic valuation of the $2k$-nacci sequence and use it to establish bounds on the solutions of the initial equation. In addition, we specify a more general family of sequences, for which we can perform a similar procedure. We also investigate a possible connection of these results with $p$-regular sequences.