Let σk(n) be the sum of the kth powers of the divisors of n. Here, we prove that if (Fn)n≥1 is the Fibonacci sequence, then the only solutions of the equation σk(Fm) = Fn in positive integers k ≥ 2, m and n have k = 2 and m ∈ {1, 2, 3}. The proof uses linear forms in two and three logarithms, lattice basis reduction, and some elementary considerations.

We show that the Conway polynomials of Fibonacci links are Fibonacci polynomials modulo 2. We deduce that, when n 6≡ 0 (mod 4) and (n, j) 6= (3, 3), the Fibonacci knot F (n) j is not a Lissajous knot. keywords: Fibonacci polynomials, Fibonacci knots, continued fractions

One-parameter Darboux deformations are established for the simple ordinary differential equation (ODE) satisfied by continuous generalizations of Fibonacci sequence recently discussed Faraoni and Atieh [Symmetry 13, 200 (2021)], who promoted a formal analogy with Friedmann in FLRW homogeneous cosmology. The method allows introduction sequences, hence Darboux-deformed (noninteger) numbers. Consi...

Let $\{F_{n}\}_{n\geq0}$ be the Fibonacci sequence. The aim of this paper is to give explicit formulae for infinite products $$\begin{equation*} \prod_{n=1}^{\infty}\left( 1+\frac{1}{F_{n}}\right) ,\quad\prod_{n=3}^{\infty}\left( 1-\frac{1}{F_{n}}\right) \end{equation*}$$ in terms values Jacobi theta functions. From we deduce algebraic independence over $\mathbf{Q}$ above numbers by applying Be...

Let us denote by Fn the n-th Fibonacci number. In this paper we show that for a fixed integer y there exists at most one exponent > 0 such Diophantine equation + Fm = ya has solution (n; m; a) in positive integers satisfying n m 0, unless 2; 3; 4; 6 or 10.

the objective of this study was to develop a new optimal parallel algorithm for matrix multiplication which could run on a fibonacci hypercube structure. most of the popular algorithms for parallel matrix multiplication can not run on fibonacci hypercube structure, therefore giving a method that can be run on all structures especially fibonacci hypercube structure is necessary for parallel matr...

In this paper, we investigate the general solution and the generalized Hyers-Ulam stability of a new functional equation satisfied by $f(x) = x^{24}$, which is called quattuorvigintic functional equation in intuitionistic fuzzy normed spaces by using the fixed point method.These results can be regarded as an important extension of stability results corresponding to functional equations on norme...

abstract. let x be a vector space over a field k of real or complex numbers.we will prove the superstability of the following golab-schinzel type equationf(x + g(x)y) = f(x)f(y); x; y 2 x;where f; g are unknown functions (satisfying some assumptions). then we generalize the superstability result for this equation with values in the field of complex numbers to the case of an arbitrary hilbert spac...

This paper will explore the relationship between the Fibonacci numbers and the Euclidean Algorithm in addition to generating a generalization of the Fibonacci Numbers. It will also look at the ratio of adjacent Fibonacci numbers and adjacent generalized Fibonacci numbers. Finally it will explore some fun applications and properties of the Fibonacci numbers.

Remark: A similar argument shows that any secondorder linear recurrent sequence also satisfies a quadratic second-order recurrence relation. A familiar example is the identity Fn−1Fn+1 − F2 n = (−1)n for Fn the nth Fibonacci number. More examples come from various classes of orthogonal polynomials, including the Chebyshev polynomials mentioned below. Second solution. We establish directly that ...