Related papers: Small Prime Powers in the Fibonacci Sequence

A basic fact taught in undergraduate algebra courses is that every finite nilpotent group is a direct product of $p$-groups. Already Bruck observed that this does not generalize to loops. In particular, there exist nilpotent loops of size…

Exponential sums with monomials are highly related to many interesting problems in number theory and well studied by many literatures. In this paper, we consider the exponential sums with polynomials and prove a new upper bound. As an…

We create a sequence version of calculus. First, we define equivalence, some fundamental operations, differential, and integral for sequences. Then, we propose sequence versions of identity function, power function, exponential function,…

Let $a > 1$. Then $a^n < n!$ for some positive integer $n$. There are several numerical sequences associated with the study of the smallest such integer which are studied in \cite{RadFact} and \cite{RadGamma}. Here we continue the…

For $k\geq 2$, the $k$-generalized Fibonacci sequence $(F_n^{(k)})_{n}$ is defined by the initial values $0,0,\ldots,0,1$ ($k$ terms) and such that each term afterwards is the sum of the $k$ preceding terms. In this paper, we search for…

We study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences. These sequences arise naturally in the study of nonlinear extensions of weighted automata, where…

We develop a general framework for finding all perfect powers in sequences derived by shifting non-degenerate quadratic Lucas-Lehmer binary recurrence sequences by a fixed integer. By combining this setup with bounds for linear forms in…

In this paper, we find all integers c having at least two representations as a difference between a Fibonacci number and a power of 2.

We present two complementary proofs that, if the lengths of $n$ sticks are sampled at random, then the probability that no $p+1$ sticks can form a $(p+1)$-sided polygon can be expressed as the product of the reciprocals of a series of terms…

In this note, we prove that for any given positive integer $\kappa$, when $n$ is bigger than a constant explicitly depending on $\kappa$, the $n$-th Fibonacci number has a primitive divisor not less than $(\kappa+1)n-1$.

We study higher-dimensional interlacing Fibonacci sequences, generated via both Chebyshev type functions and $m$-dimensional recurrence relations. For each integer $m$, there exist both rational and integer versions of these sequences,…

This paper introduces a new method to find the next prime number after a given prime ${P}$. The proposed method is used to derive a system of inequalities, that serve as constraints which should be satisfied by all primes whose successor is…

The $n$-th Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube $Q_n$ induced by binary strings with no two consecutive ones. We determine $\pi(\Gamma_n) = 2^n$ for $n \le 6$, so the pebbling number of $\Gamma_n$ equals that of the…

Double Fibonacci sequences are introduced and they are related to operations with Fibonacci modules. Generalizations and examples are also discussed.

1. There is no existing any quadratic interval $\eta_{n}:=(n^{2},(n+1)^{2}],$ which contains less than 2 prime numbers. The number of prime numbers within $\eta_{n}$ goes averagely linear with n to infinity. 2. The exact law of the number…

We apply the Desnanot-Jacobi identity to give an alternative proof of the determinants whose entries are rising powers of the Fibonacci numbers given by Prodinger. We then generalize the determinants to include entries that are rising…

The Fibonacci sequence is periodic modulo every positive integer $m>1$, and perhaps more surprisingly, each period has exactly 1, 2, or 4 zeros that are evenly spaced, which also holds true for more general $K$-Fibonacci sequences. This…

This note presents an analytic technique for proving the linear independence of certain small subsets of real numbers over the rational numbers. The applications of this test produce simple linear independence proofs for the subsets of…

In this paper, we prove that there is no x>=4 such that the difference of x-th powers of two consecutive Fibonacci numbers greater than 0 is a Lucas number.

We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer…