16 research outputs found
On a problem of Pillai with k-generalized Fibonacci numbers and powers of 2
For an integer , let be the --generalized Fibonacci sequence which starts with ( terms) and each term afterwards is the sum of the preceding terms. In this paper, we find all integers having at least two presentations as a difference between a --generalized Fibonacci number and a powers of 2 for any fixed . This paper extends previous work from [9] for the case and [6] for the case
On Pillai’s problem with Pell numbers and powers of 2
In this paper, we find all integers c having at least two representations as a difference between a Pell number and a power of 2
On the Problem of Pillai with Fibonacci numbers and powers of
International audienceConsider 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 Pillai’s problem with X-coordinates of Pell equations and powers of 2 II
In this paper, we show that if (X_n, Y_n) is the nth solution of the Pell equation X^2−dY^2= ±1 for some non-square d, then given any int eger c, the equation c = X_n−2^m has at most 2 integer solutions (n, m)withn ≥ 0andm ≥ 0, except for the only pair (c, d)=(−1, 2). Moreover, we show that this bound is optimal. Additionally, we propose a conjecture about the number of solutions of Pillai’s problem in linear recurrent sequences
On the multiplicity in Pillai\u27s problem with Fibonacci numbers and powers of a fixed prime
Let ( {F_n}_{ngeq 0} ) be the sequence of Fibonacci numbers and let (p) be a prime. For an integer (c) we write (m_{F,p}(c)) for the number of distinct representations of (c) as (F_k-p^ell) with (kge 2) and (ellge 0). We prove that (m_{F,p}(c)le 4)