A Pillai-Catalan-type problem involving Fibonacci numbers

Abstract

This paper addresses A Pillai-Catalan-type problem assosiated with Fibonacci numbers. Let F-n be the Fibonacci numbers defined by the recurrence relation F-1 = 1, F-2 = 1 and F-n = Fn-1 + Fn-2 for all n >= 3. We will find all positive integer solution to the equation 3(x) - F(n)2(y) = 1 using properties of Fibonacci numbers, linear forms of logarithms and Baker-Davenport reduction method.