A Pillai-Catalan-type problem involving Fibonacci numbers

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发表日期：

2024年5月5日

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Seyran S. Ibrahimov,
Nazim I. Mahmudov

摘要

This paper addresses A Pillai-Catalan-type problem assosiated with Fibonacci numbers. Let $F_{n}$ $F_{n}$ be the Fibonacci numbers defined by the recurrence relation $F_{1}=1$ $F_{1} = 1$ , $F_{2}=1$ $F_{2} = 1$ and $F_{n}=F_{n-1}+F_{n-2}$ $F_{n} = F_{n - 1} + F_{n - 2}$ for all $n \ge 3$ $n \geq 3$ . We will find all positive integer solution to the equation $3^{x}-F_{n}2^{y}=1$ $3^{x} - F_{n} 2^{y} = 1$ using properties of Fibonacci numbers, linear forms of logarithms and Baker-Davenport reduction method.