On a problem of Pillai with k-generalized Fibonacci numbers and powers of 2

For an integer $k\geq 2$, let $\{F^{(k)}_{n} \}_{n\geq 0}$ be the $k$--generalized Fibonacci sequence which starts with $0, \ldots, 0, 1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all integers $c$ having at least two presentations as a difference between a $k$--generalized Fibonacci number and a powers of 2 for any fixed $k \geqslant 4$. This paper extends previous work from [9] for the case $k=2$ and [6] for the case $k=3$

On the Diophantine equation Un−bm=cU_n-b^m = c

Let $(U_n)_{n\in \mathbb{N}}$ be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants $B$ and $N_0$ such that for any $b,c\in \mathbb{Z}$ with $b> B$ the equation $U_n - b^m = c$ has at most two distinct solutions $(n,m)\in \mathbb{N}^2$ with $n\geq N_0$ and $m\geq 1$. Moreover, we apply our result to the special case of Tribonacci numbers given by $T_1= T_2=1$, $T_3=2$ and $T_{n}=T_{n-1}+T_{n-2}+T_{n-3}$ for $n\geq 4$. By means of the LLL-algorithm and continued fraction reduction we are able to prove $N_0=1.1\cdot 10^{37}$ and $B=e^{438}$. The corresponding reduction algorithm is implemented in Sage.Comment: 34 page

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 33

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 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)

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