# On the problem of Pillai with $k$--generalized Fibonacci numbers and powers of $3$

## TL;DR

This paper investigates the representations of integers as differences between $k$-generalized Fibonacci numbers and powers of 3, extending previous work on Fibonacci and Tribonacci numbers to a broader class.

## Contribution

It characterizes all integers with multiple representations as such differences for generalized Fibonacci sequences, generalizing prior results.

## Key findings

- Identifies all integers with at least two representations as differences.
- Extends previous results from Fibonacci and Tribonacci sequences.
- Provides a complete classification for $k$-generalized Fibonacci differences.

## Abstract

For an integer $k\ge 2$, let $\{F^{(k)}_{n}\}_{n\ge 2-k}$ be the $ k$--generalized Fibonacci sequence which starts with $0, \ldots, 0,1$ (a total of $k$ terms) and for which each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all integers $ c $ with at least two representations as a difference between a $ k $-generalized Fibonacci number and a power of $ 3 $. This paper continues the previous work of the first author for the Fibonacci numbers, and the Tribonacci numbers.