A note on Q-matrices and higher order Fibonacci polynomials

The results described in a recent article, relative to a representation formula for the generalized Fibonacci sequences in terms of Q-matrices are extended to the case of Fibonacci, Tribonacci and R-bonacci polynomials.

Representation of integers by k-generalized Fibonacci sequences and applications in cryptography

The aim of this paper is to construct a relation between tribonacci numbers and generalized tribonacci numbers. Besides, certain conditions are obtained to generalize the representation of a positive integer [Formula: see text] which is determined in [S. Badidja and A. Boudaoud, Representation of positive integers as a sum of distinct tribonacci numbers, J. Math. Statistic. 13 (2017) 57–61] for a [Formula: see text]-generalized Fibonacci numbers [Formula: see text]. Lastly, some applications to cryptography are given by using [Formula: see text].

On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes

The k-generalized Fibonacci sequence ( F n ( k ) ) n (sometimes also called k-bonacci or k-step Fibonacci sequence), with k ≥ 2 , is defined by the values 0 , 0 , … , 0 , 1 of starting k its terms and such way that each term afterwards is the sum of the k preceding terms. This paper is devoted to the proof of the fact that the Diophantine equation F m ( k ) = m t , with t > 1 and m > k + 1 , has only solutions F 12 ( 2 ) = 12 2 and F 9 ( 3 ) = 9 2 .

Diophantine quadruples with values in k-generalized Fibonacci numbers

We consider for integersk≥ 2 thek–generalized Fibonacci sequencesF(k):=$\begin{array}{} (F_n^{(k)})_{n\geq 2-k}, \end{array} $whose firstkterms are 0, …, 0, 1 and each term afterwards is the sum of the precedingkterms. In this paper, we show that there does not exist a quadruple of positive integersa1<a2<a3<a4such thataiaj+ 1 (i≠j) are all members ofF(k).

On prime factors of the sum of two k-Fibonacci numbers

We consider for integers [Formula: see text] the [Formula: see text]-generalized Fibonacci sequences [Formula: see text], whose first [Formula: see text] terms are [Formula: see text] and each term afterwards is the sum of the preceding [Formula: see text] terms. We give a lower bound for the largest prime factor of the sum of two terms in [Formula: see text]. As a consequence of our main result, for every fixed finite set of primes [Formula: see text], there are only finitely many positive integers [Formula: see text] and [Formula: see text]-integers which are a non-trivial sum of two [Formula: see text]-Fibonacci numbers, and all these are effectively computable.