# On the problem of Pillai with Fibonacci numbers, Padovan numbers, and Tribonacci numbers and powers of $3$

## TL;DR

This paper completely characterizes all integers that can be expressed in at least two ways as the difference between Fibonacci, Padovan, or Tribonacci numbers and powers of 3.

## Contribution

It provides a complete solution to the problem of finding multiple representations of integers as differences involving these special sequences and powers of 3.

## Key findings

- Identifies all integers with multiple representations in the specified forms.
- Establishes the uniqueness or finiteness of such representations.
- Extends the classical Pillai problem to these sequences and powers of 3.

## Abstract

Consider the sequences: $ \{F_{n}\}_{n\geq 0} $ of Fibonacci numbers defined by $ F_0=0 $, $ F_1 =1$ and $ F_{n+2}=F_{n+1}+ F_{n} $ for all $ n\geq 0 $; $ \{P_{n}\}_{n\geq 0} $ of Padovan numbers defined by $ P_0=0 $, $ P_1 =1 = P_2 $ and $ P_{n+3}=P_{n+1}+ P_{n} $ for all $ n\geq 0 $; and $ \{T_{n}\}_{n\geq 0} $ of Tribonacci numbers defined by $ T_0=0 $, $ T_1 =1= T_2$ and $ T_{n+3}=T_{n_2}+T_{n+1}+ T_{n} $ for all $ n\geq 0 $. In this paper, we find all integers $ c $ having at least two representations as a difference between: a Fibonacci number and a power of $ 3 $; a Padovan number and a power of $3$; and a Tribonacci number and a power of $3$.