On a problem of Pillai with Fibonacci numbers and powers of 3

Abstract

Consider the sequence {Fn} n≥0 of Fibonacci numbers defined by F 0 = 0, F 1 = 1 and F n+2 = F n+1 + Fn for all n ≥ 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.

On the problem of Pillai with Padovan numbers and powers of 3

“…Since then, different variations of the Pillai equation have been studied. Some recent results for the different variations of the Pillai problem involving Fibonacci numbers, Tribonacci numbers, Pell numbers, the k-generalized Fibonacci numbers and other generalized linearly recurrent sequences, with powers of 2, have been completely studied, for example, in [3,4,5,6,7,8,9,10,11,12].…”

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

“…It was proved in [14] that the contribution of the roots which are inside the unit circle to the formula (10) is very small, namely that the approximation ( 11)…”

“…In the past years, several special cases of the problem of Pillai have been studied. See, for example, [4,7,8,10,11,12,16,17].…”

“…with (n, m) = (n 1 , m 1 ). The cases k = 2 and k = 3 have been solved completely by the first author in [10] and [11], respectively. So, we focus on the case k 4.…”

An approach to Pillai's problem with the Pell sequence and the powers of 3

“…For example in [13], authors shows that the set of integers admitting at least two representations of the form F n − 2 a is {−30, −11, −3, −1, 0, 1, 5, 85}. Since then, many studies including integer sequences were handled by the authors (see [3,[6][7][8][9][10][11][12][15][16][17][18][19]). We consider Pell sequence which is defined by P 0 = 0, P 1 = 1 and P n = 2P n−1 + P n−2 .…”