On Pillai's problem involving two linear recurrent sequences : Padovan and Fibonacci

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

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…

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

In this paper, we find all integers c having at least two representations as a difference between a Fibonacci number and a power of 2.

On the multiplicity in Pillai's problem with Fibonacci numbers and powers of a fixed prime

Let $ \{F_n\}_{n\ge 0} $ be the sequence of Fibonacci numbers and let $p$ be a prime. For an integer $c$ we write $m_{F,p}(c)$ for the number of distinct representations of $c$ as $F_k-p^\ell$ with $k\ge 2$ and $\ell\ge 0$. We prove that $m_{F,p}(c)\le 4$.

一类涉及斐波那契数的Pillai-Catalan型问题

A Pillai-Catalan-type problem involving Fibonacci numbers

本文针对涉及斐波那契数的Pillai-Catalan型问题，结合斐波那契数性质、对数线性型与Baker-Davenport约化法，求解不定方程$3^x - F_n 2^y = 1$的所有正整数解。

On a problem of Pillai with $k$-generalised Fibonacci numbers and powers of $2$

For an integer $ k\geq 2 $, let $ \{F^{(k)}_{n} \}_{n\geq 0}$ be the $ k$--generalized Fibonacci sequence which starts with $ 0, \ldots, 0, 1 $ ($ k $ terms) and each term afterwards is the sum of the $ k $ preceding terms. In this paper, we find all integers $c$ having at least two representations as a difference between a $k$--generalized Fibonacci number and a powers of 2 for any fixed $k \geqslant 4$. This paper extends previous work from [9] for the case $k=2$ and [6] for the case $k=3$.

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

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.