Abstract
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 representations 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 $k=3$.
Highlights
The problem of Pillai states that for each fixed integer c ≥ 1, the Diophantine equation ax − by = c, min{x, y} ≥ 2, (1.1)has only a finite number of positive solutions {a, b, x, y}
We find all integers c with at least two representations as a difference between a k-generalized Fibonacci number and a power of 3
In 1936, in the special case (a, b) = (3, 2) which is a continuation of the work of Herschfeld [19, 20] in 1935, Pillai conjectured that the only integers c admitting at least two representations of the form 2x − 3y are given by
Summary
Has only a finite number of positive solutions {a, b, x, y} This problem is still open; the case c = 1 is the conjecture of Catalan and was proved by Mihailescu [22]. The generalized Fibonacci analogue of the problem of Pillai under the same conditions as in (1.1) concerns studying for fixed (k, ) all values of the integer c such that the equation. We study a related problem and we find all integers c admitting at least two representations of the form Fn(k) − 3m for some positive integers k, n, and m This can be interpreted as solving the equation.
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