On the Fibonacci numbers and their sums which are close to a power of 3

Abstract

Let (L_n)_{n ≥ 0} be the Lucas sequence defined by L_n+2 = L_n+1 + L_n for all n ≥ 0, with initial values L₀ = 2 and L₁ = 1. In this paper, we find all the Fibonacci numbers 2F_n and sums of two Fibonacci numbers which are close to a power of 3. As a corollary, we determine all Lucas numbers close to a power of 3. To prove our results, we will use Baker's theory lower bound for linear forms in logarithms of algebraic numbers, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.