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

Abstract

Consider the sequences: $\{F_n{\}}_{n\ge 0}$ of Fibonacci numbers defined by $F_0=0$, $F_1=1$ and $F_{n+2}=F_{n+1}+F_n$ for all $n\ge 0$; $\{P_n{\}}_{n\ge 0}$ of Padovan numbers defined by $P_0=0$, $P_1=1=P_2$ and $P_{n+3}=P_{n+1}+P_n$ for all $n\ge 0$; and $\{T_n{\}}_{n\ge 0}$ 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 $n\ge 0$. 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 $3$.

34 pages: 3 papers are combined into one single submission for the benefit of the reader

References in corpus (1)

• On a problem of Pillai with Fibonacci numbers and powers of 2