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In this paper, we present a new encryption and decryption algorithm that includes Fibonacci numbers, linear mappings, and change of basis.

In his fascinating book Wonders of Numbers, Clifford Pickover introduces the Ana sequence and fractal, two self-referential constructions arising from the use of language. This paper answers Pickover's questions on the relative composition of sequence terms and the dimension of the fractal. In the process, it introduces a novel way of obtaining fractals from iterative set operations. Also, it presents a beautiful variant of the Ana constructions involving the golden ratio. In conclusion, it suggests ways of constructing similar fractals for the Morse-Thue and "Look and Say" sequences.

Starting from divisibility problem for Fibonacci numbers, we introduce Fibonacci divisors, related hierarchy of Golden derivatives in powers of the Golden Ratio and develop corresponding quantum calculus. By this calculus, the infinite hierarchy of Golden quantum oscillators with integer spectrum determined by Fibonacci divisors, the hierarchy of Golden coherent states and related Fock–Bargman representations in space of complex analytic functions are derived. It is shown that Fibonacci divisors with even and odd $k$ describe Golden deformed bosonic and fermionic quantum oscillators, correspondingly. By the set of translation operators we find the hierarchy of Golden binomials and related Golden analytic functions, conjugate to Fibonacci number $F_k$. In the limit $k\to 0$, Golden analytic functions reduce to classical holomorphic functions and quantum calculus of Fibonacci divisors to the usual one. Several applications of the calculus to quantum deformation of bosonic and fermionic oscillator algebras, $R$-matrices, geometry of hydrodynamic images and quantum computations are discussed.

The set of maximally fermion–boson entangled Bell super-coherent states is introduced. A superposition of these states with separable bosonic coherent states, represented by points on the super-Bloch sphere, we call the Bell-based super-coherent states. Entanglement of bosonic and fermionic degrees of freedom in these states is studied by using displacement bosonic operator. It acts on the super-qubit reference state, representing superposition of the zero and the one super-number states, forming computational basis super-states. We show that the states are completely characterized by displaced Fock states, as a superposition with non-classical, the photon added coherent states, and the entanglement is independent of coherent state parameter $\alpha$ and of the time evolution. In contrast to never orthogonal Glauber coherent states, our entangled super-coherent states can be orthogonal. The uncertainty relation in the states is monotonically growing function of the concurrence and for entangled states we get non-classical quadrature squeezing and representation of uncertainty by ratio of two Fibonacci numbers. The sequence of concurrences, and corresponding uncertainties $\hslash F_n/F_{n+1}$, in the limit $n\to \infty$, convergent to the Golden ratio uncertainty $\hslash /\phi$, where $\phi =\frac{1+\sqrt{5}}{2}$ is found.

A set $\{a_1,a_2,\dots ,a_m\}$ of positive integers is called a Diophantine $m$-tuple if $a_i{a}_j+1$ is a perfect square for all $1\le i<j\le m$. Let $\{a,b,c\}$ be the Diophantine triple with $c>\max \{a,b\}$. In this paper, we find the condition for the reduction of third element $c$, and using this result, we prove the extendibility of Diophantine pair $\{F_{k-1}{F}_{k+1},F_{k-2}{F}_{k+2}\}$, where $F_n$ is the $n$-th Fibonacci number.

Let z(N) be the order of appearance of N in the Fibonacci sequence. This is the smallest positive integer k such that N divides the kth Fibonacci number. We show that each of the six total possible orderings among z(N), z(N + 1), z(N + 2) appears infinitely often. We also show that for each nonzero even integer c and many odd integers c the equation z(N) = z(N + c) has infinitely many solutions N, but the set of solutions has asymptotic density zero. The proofs use a result of Corvaja and Zannier on the height of a rational function at 𝒮-unit points as well as sieve methods.

In this paper, we show that if p ≡ 1 (mod 4) is prime, then F_p admits a representation of the form u² + pv² for some integers u and v, where F_n is the nth Fibonacci number.

We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalizing known results for Catalan numbers.

We consider equations of the form $F_n=P(x)$, where $P$ is a polynomial with integral coefficients and $F_n$ is the $n$th Fibonacci number that is, $F_0=0,F_1=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n>1.$ In particular, for each $k\in {\mathbb{N}}_{+}$, we prove the existence of a polynomial $F_k\in \mathbb{Z}[x]$ of degree $2k-1$ such that the Diophantine equation $F_k(x)=F_m$ has infinitely many solutions in positive integers $(x,m)$. Moreover, we present results of our numerical search concerning the existence of even degree polynomials representing many Fibonacci numbers. We also determine all integral solutions $(n,x)$ of the Diophantine equations $\left(\genfrac{}{}{0.0pt}{}{x}{2}\right)+d=F_n$ for $-20\le d\le 20$ and $F_n=\left(\genfrac{}{}{0.0pt}{}{x}{5}\right)$.

In this paper, we prove that the set of integers which can be represented as the sum of a Fibonacci number and a prime in at least one and at most $37$ ways has the lower asymptotic density at least $0.143$.

In this paper, we show that there is at most one value of the positive integer $X$ participating in the Pell equation $X^2-d{Y}^2=\pm 1$, which is a difference of two Fibonacci numbers.

In this paper, we show that each non-degenerate Lucas sequence contains only finitely many terms which can be written as products of double factorials or as sums or differences of double factorials. In particular, we find all the solutions to the products of double factorials which are sums or differences of two Fibonacci numbers, and Fibonacci numbers which are sums or differences of two double factorials.

In this paper, we consider the series of Fibonacci $n$-step numbers $F_k^{(n)}$. Thereby, we establish some properties and theorems of these numbers for $k=0,\pm 1,\pm 2,\dots$ and $n>1$.

In this paper, we deal with the periodicity of solutions of the following general system rational of difference equations:

In this paper, we define (m, t)-extension of the Fibonacci p-numbers and Golden (p, m, t)-proportions, where p ≥ 0 is integer, m > 0, and t > 0. We establish a relation among golden (p, m, t)-proportion, golden (p, m)-proportion and golden p-proportion. Thereby, we define a new Fibonacci G_{p, m, t} matrix. Then we show that by proper selection of the initial terms for the (m, t)-extension of the Fibonacci p-numbers, we can apply Fibonacci coding/decoding in G_{p, m, t} matrix. Also it is obvious that for t = 1, the relations among the code elements for all values of p (non-negative integer) and m (> 0) coincide with the relations among the code matrix elements for all values of p and m (> 0) with the same initial terms (see the paper coding theory on the m-extension of the Fibonacci p-numbers, Chaos, Solitons and Fractals42 (2009) 2522–2530).

In this paper, we introduce h(x) Fibonacci p-numbers polynomials and a new Fibonacci Q_p,h(x) matrix for h(x) Fibonacci p-numbers polynomials where p(> 0) is integer and h(x)(> 0) is a polynomial with real coefficients. Thereby, we discuss various properties of Q_p,h(x) matrix and the coding theory followed from the Q_p,h(x) matrix. We establish the relations among the code matrix elements for p(> 0) is integer and h(x)(> 0). We also show that the relations, among the code matrix elements for h(x) = m (> 0), coincide with the relations among the code matrix elements for coding theory on the m-extension of the Fibonacci p-numbers. The relations among the code matrix elements for h(x) = 1, coincide with the generalized relations among the code matrix elements for Fibonacci coding theory. In general, correct ability of this method, which is independent of h(x), increases as p increases but h(x) being a polynomial, improves the cryptography protection.

In this paper, we consider the series of Tribonacci numbers. Thereby, we introduce a new coding theory called Tribonacci coding theory based on Tribonacci numbers and show that in the simplest case, the correct ability of this method is 99.80% whereas the correct ability of the Fibonacci coding/decoding method is 93.33%.

In this paper, we develop hybrid matrix cryptography followed by Fibonacci Q_p,h(x) matrix, where h(x) is a polynomial with real coefficients and p = 1, 2, 3, …. They are the generalization of classical Fibonacci Q matrix for a continuous domain. They are used for designing super reliable cryptosystems.

We consider the series of Fibonacci n-step numbers and a class of square matrix of order n based on Fibonacci n-step numbers with determinant +1 or -1. Thereby, we introduce a new coding theory called Fibonacci n-step coding theory and establish generalized relation among the code elements for all values of n. For n = 2, the correct ability of the method is 93.33%; for n = 3, the correct ability of the method is 99.80% and for n = 4, the correct ability of the method is 99.998%. In general, the correct ability of the method increases as n increases.

In this paper, we introduced h(x) extension of m sequences of Fibonacci numbers polynomials of order m and a new S_n matrix of order m where h(x)(> 0) is a polynomial with real coefficients. Thereby, we discuss various properties of S_n matrix and the Fibonacci coding theory followed from this matrix. We established the relations among the code matrix elements, error deduction and correction for this coding theory. In general, correct ability potential of this method increases and it is independent of h(x). But h(x) being a polynomial, improves the cryptography protection.