Repdigits as difference of two Fibonacci or Lucas numbers

In the present study we investigate all repdigits which are expressed as a difference of two Fibonacci or Lucas numbers. We show that if $F_{n}-F_{m}$ is a repdigit, where $F_{n}$ denotes the $n$-th Fibonacci number, then $(n,m)\in \{(7,3),(9,1),(9,2),(11,1),(11,2),$ $(11,9),(12,11),(15,10)\}.$ Further, if $L_{n}$ denotes the $n$-th Lucas number, then $L_{n}-L_{m}$ is a repdigit for $(n,m)\in\{(6,4),(7,4),(7,6),(8,2)\},$ where $n>m.$Namely, the only repdigits that can be expressed as difference of two Fibonacci numbers are $11,33,55,88$ and $555$; their representations are $11=F_{7}-F_{3},\33=F_{9}-F_{1}=F_{9}-F_{2},\55=F_{11}-F_{9}=F_{12}-F_{11},\88=F_{11}-F_{1}=F_{11}-F_{2},\555=F_{15}-F_{10}$ (Theorem 2). Similar result for difference of two Lucas numbers: The only repdigits that can be expressed as difference of two Lucas numbers are $11,22$ and $44;$ their representations are $11=L_{6}-L_{4}=L_{7}-L_{6},\ 22=L_{7}-L_{4},\4=L_{8}-L_{2}$ (Theorem 3).

The Proof of a Conjecture Related to Divisibility Properties of z(n)

The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n ≥ 1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k ≥ 1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture).

Some generalized results related to Fibonacci sequence

Cassini's identity states that for the nth Fibonacci number Fn+1Fn-1-Fn2=(-1)n. We generalize Fibonacci sequence in terms of the number of sequences. Fibonacci sequence is the particular case of generating only one sequence. This generalization is used to generalize Cassini’s identity. Moreover we prove few more results which can be seen as generalized form of the results which hold for Fibonacci sequence.

The Pyramid – Resonator of waves

The Great Pyramid of Giza has fascinated us all as it encodes enormous amount of numerical coincidences such as dimensional precision, movement of our planet, speed of light, the golden ratio of Pi & Phi, etc.Studies have reasoned that the great pyramid of Giza has expressed the key ratio of an AC voltage sine wave as well as the ratios of Fibonacci number in developing the pyramidal design. Therefore in this study, the pyramid structure is considered as a resonator of waves where reflection of waves is an obvious phenomenon. The waves entering the pyramidal resonator will be reflected inward as they reflect from a curved surface according to the law of reflection. Since, a reflecting wave involves the energy-transport process, it determines our main objective to review and internalize the energy caused by reflection of the waves which occurs inside the pyramidal resonator. It is assumed that there is a strength point of such energy due to a higher volume of reflected waves to a single point. According to the law of reflection, when reflection occurs through a curved surface, it focuses incoming parallel waves to a convergence spot. This project is subjected to study the pyramid as a resonator of waves and aims to detect, observationally, the strength point of energy assumed to be caused by maximum number of reflected waves.

The Pyramid – Resonator of waves

The Great Pyramid of Giza has fascinated us all as it encodes enormous amount of numerical coincidences such as dimensional precision, movement of our planet, speed of light, the golden ratio of Pi & Phi, etc.Studies have reasoned that the great pyramid of Giza has expressed the key ratio of an AC voltage sine wave as well as the ratios of Fibonacci number in developing the pyramidal design. Therefore in this study, the pyramid structure is considered as a resonator of waves where reflection of waves is an obvious phenomenon. The waves entering the pyramidal resonator will be reflected inward as they reflect from a curved surface according to the law of reflection. Since, a reflecting wave involves the energy-transport process, it determines our main objective to review and internalize the energy caused by reflection of the waves which occurs inside the pyramidal resonator. It is assumed that there is a strength point of such energy due to a higher volume of reflected waves to a single point. According to the law of reflection, when reflection occurs through a curved surface, it focuses incoming parallel waves to a convergence spot. This project is subjected to study the pyramid as a resonator of waves and aims to detect, observationally, the strength point of energy assumed to be caused by maximum number of reflected waves.

On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers

Let (tn(r))n≥0 be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence tn(r)=tn−1(r)+⋯+tn−r(r) for n≥r, with initial values t0(r)=0 and ti(r)=1, for all 1≤i≤r. In 2002, Grossman and Luca searched for terms of the sequence (tn(2))n, which are expressible as a sum of factorials. In this paper, we continue this program by proving that, for any ℓ≥1, there exists an effectively computable constant C=C(ℓ)>0 (only depending on ℓ), such that, if (m,n,r) is a solution of tm(r)=n!+(n+1)!+⋯+(n+ℓ)!, with r even, then max{m,n,r}<C. As an application, we solve the previous equation for all 1≤ℓ≤5.

Phyllotactic patterning of gerbera flower heads

Phyllotaxis, the distribution of organs such as leaves and flowers on their support, is a key attribute of plant architecture. The geometric regularity of phyllotaxis has attracted multidisciplinary interest for centuries, resulting in an understanding of the patterns in the model plants Arabidopsis and tomato down to the molecular level. Nevertheless, the iconic example of phyllotaxis, the arrangement of individual florets into spirals in the heads of the daisy family of plants (Asteraceae), has not been fully explained. We integrate experimental data and computational models to explain phyllotaxis in Gerbera hybrida. We show that phyllotactic patterning in gerbera is governed by changes in the size of the morphogenetically active zone coordinated with the growth of the head. The dynamics of these changes divides the patterning process into three phases: the development of an approximately circular pattern with a Fibonacci number of primordia near the head rim, its gradual transition to a zigzag pattern, and the development of a spiral pattern that fills the head on the template of this zigzag pattern. Fibonacci spiral numbers arise due to the intercalary insertion and lateral displacement of incipient primordia in the first phase. Our results demonstrate the essential role of the growth and active zone dynamics in the patterning of flower heads.

Quantum calculus of Fibonacci divisors and infinite hierarchy of Bosonic–Fermionic Golden quantum oscillators

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 [Formula: see text] 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 [Formula: see text]. In the limit [Formula: see text], 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, [Formula: see text]-matrices, geometry of hydrodynamic images and quantum computations are discussed.

The sum of a prime and a Fibonacci number

In this paper, we show that the lower density of integers representable as the sum of a prime and a Fibonacci number is at least [Formula: see text].

SUPER FIBONACCI GRACEFUL LABELING FOR GENERALIZED (a,m)-SHELL GRAPH AND MERGED WITH SOME GRAPHS

A graph $G$ with $p$ vertices and $q$ edges has super Fibonacci graceful labeling if there exists an injective map $f : V(G) \rightarrow \left\{F_{0}, F_{1},F_{2},\ldots F_{q}\right\}$ where $F_{k}$ is the $k^{th}$ Fibonacci number of the Fibonacci series such that its induced map $f^{+}: E(G) \rightarrow\left\{F_{1},F_{2},F_{3},\ldots F_{q}\right\}$ defined by $f^{+}(xy)$ =$\left|f(x) - f(y)\right|$ $\forall$ $xy \in G,$ is a bijective map. In this paper, we investigate the existence of super Fibonacci graceful labeling for the various types of $(a, m)$ - shell graph and $(a, m)$ - shell graph merged with some graphs.