On a variant of Pillai's problem II

Abstract

Abstract. In this paper, we show that there are only finitely many c such that the equation Un −Vm = c has at least two distinct solutions (n, m), where {Un} n 0 and {Vm} m 0 are given linear recurrence sequences.

Pillai’s problem with the Fibonacci andPadovan sequences

“…In the same way, ̸ = 1 and 1 ̸ = 1. With this conventions, our result is the following: (18,10,17,9), (18,11,8,6), (18,11,5,5), (18,11,0,4), (19,11,14,4), (19,12,7,9), (20,11,17,4), (20,12,14,8), (20,12,11,5), (20,12,10,3), (20,12,9,0), (20,13,9,11), (20,14,9,13), (21,…”

“…Both results are ineffective. In [16], Pillai conjectured that in the case ( , ) = (2, 3) the only solutions of equation 1.1are (3, 2, 1, 1), (5,3,3,1) and (8,5,4,1). This conjecture remained open for about 37 years and was confirmed in [20] by Stroeker and Tijdeman by using Baker's theory on linear forms in logarithms.…”

“…Many other cases have been studied, see for example [3,6,7,10,12,13]. In [5], there is a general result, namely that if U and V satisfy some natural conditions, then equation (12) has only finitely many solutions which furthermore are all effectively computable. We recall that the Fibonacci sequence ( ) 0 is given by 0 = 0, 1 = 1 and the recurrence formula…”

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

“…Since then, different variants of the Pillai equation have been studied. Some recent results for the different variants of the Pillai problem involving Fibonacci numbers, Tribonacci numbers, Pell numbers and the k-generalized Fibonacci numbers with powers of 2 have been intensively studied in [3][4][5][6][7]10,11].…”

On Pillai's Problem With Tribonacci Numbers and Powers of 2

“…We note that in the recent paper [4], the authors have studied the same problem but involving both {F n } n≥0 and {T m } m≥0 . That is, they found all integers c having two nontrivial representations as F n − T m for some positive integers m and n.…”