scholarly journals On the l.c.m. of shifted Fibonacci numbers

2021 ◽  
pp. 1-10
Author(s):  
Carlo Sanna
Keyword(s):  
Rational Number ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Random Variables ◽  
Periodic Sequence ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Number Formula ◽  
Dilogarithm Function

Let [Formula: see text] be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that [Formula: see text] where [Formula: see text] is the least common multiple and [Formula: see text] is the golden ratio. We prove that for every periodic sequence [Formula: see text] in [Formula: see text] there exists an effectively computable rational number [Formula: see text] such that [Formula: see text] Moreover, we show that if [Formula: see text] is a sequence of independent uniformly distributed random variables in [Formula: see text] then [Formula: see text] where [Formula: see text] is the dilogarithm function.

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

2021 ◽  
pp. 2150075
Author(s):  
Oktay K. Pashaev
Keyword(s):  
Analytic Functions ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Holomorphic Functions ◽  
Quantum Calculus ◽  
Infinite Hierarchy ◽  
Translation Operators ◽  
Number Formula ◽  
Divisibility Problem

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.

scholarly journals On primitive covering numbers

2016 ◽  
Vol 13 (01) ◽  
pp. 27-37
Author(s):  
Lenny Jones ◽  
Daniel White
Keyword(s):  
Positive Integer ◽  
Covering Number ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Exact Number ◽  
Covering Numbers ◽  
Covering System ◽  
Number Formula ◽  
Infinite Set

In 2007, Zhi-Wei Sun defined a covering number to be a positive integer [Formula: see text] such that there exists a covering system of the integers where the moduli are distinct divisors of [Formula: see text] greater than 1. A covering number [Formula: see text] is called primitive if no proper divisor of [Formula: see text] is a covering number. Sun constructed an infinite set [Formula: see text] of primitive covering numbers, and he conjectured that every primitive covering number must satisfy a certain condition. In this paper, for a given [Formula: see text], we derive a formula that gives the exact number of coverings that have [Formula: see text] as the least common multiple of the set [Formula: see text] of moduli, under certain restrictions on [Formula: see text]. Additionally, we disprove Sun’s conjecture by constructing an infinite set of primitive covering numbers that do not satisfy his primitive covering number condition.

scholarly journals On the least common multiple of random q-integers

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Probabilistic Model ◽  
Expected Value ◽  
Random Set ◽  
Common Multiple ◽  
Least Common Multiple

AbstractFor every positive integer n and for every $$\alpha \in [0, 1]$$ α ∈ [ 0 , 1 ] , let $${\mathcal {B}}(n, \alpha )$$ B ( n , α ) denote the probabilistic model in which a random set $${\mathcal {A}} \subseteq \{1, \ldots , n\}$$ A ⊆ { 1 , … , n } is constructed by picking independently each element of $$\{1, \ldots , n\}$$ { 1 , … , n } with probability $$\alpha $$ α . Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $${\mathcal {A}}$$ A .Let q be an indeterminate and let $$[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in {\mathbb {Z}}[q]$$ [ k ] q : = 1 + q + q 2 + ⋯ + q k - 1 ∈ Z [ q ] be the q-analog of the positive integer k. We determine the expected value and the variance of $$X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )$$ X : = deg lcm ( [ A ] q ) , where $$[{\mathcal {A}}]_q := \big \{[k]_q : k \in {\mathcal {A}}\big \}$$ [ A ] q : = { [ k ] q : k ∈ A } . Then we prove an almost sure asymptotic formula for X, which is a q-analog of the result of Cilleruelo et al.

scholarly journals Metaplectic flavor symmetries from magnetized tori

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Yahya Almumin ◽  
Mu-Chun Chen ◽  
Víctor Knapp-Pérez ◽  
Saúl Ramos-Sánchez ◽  
Michael Ratz ◽  
...  
Keyword(s):  
Zero Mode ◽  
Metaplectic Groups ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Yukawa Couplings ◽  
Flavor Symmetries ◽  
Euler’S Theorem ◽  
Modular Transformations ◽  
Analytic Expressions ◽  
Geometric Picture

Abstract We revisit the flavor symmetries arising from compactifications on tori with magnetic background fluxes. Using Euler’s Theorem, we derive closed form analytic expressions for the Yukawa couplings that are valid for arbitrary flux parameters. We discuss the modular transformations for even and odd units of magnetic flux, M, and show that they give rise to finite metaplectic groups the order of which is determined by the least common multiple of the number of zero-mode flavors involved. Unlike in models in which modular flavor symmetries are postulated, in this approach they derive from an underlying torus. This allows us to retain control over parameters, such as those governing the kinetic terms, that are free in the bottom-up approach, thus leading to an increased predictivity. In addition, the geometric picture allows us to understand the relative suppression of Yukawa couplings from their localization properties in the compact space. We also comment on the role supersymmetry plays in these constructions, and outline a path towards non-supersymmetric models with modular flavor symmetries.

scholarly journals On Generalization of Fibonacci, Lucas and Mulatu Numbers

2020 ◽  
Vol 1 (3) ◽  
pp. 112-122
Author(s):  
Agung Prabowo
Keyword(s):  
Continued Fraction ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Binet Formula

Fibonacci numbers, Lucas numbers and Mulatu numbers are built in the same method. The three numbers differ in the first term, while the second term is entirely the same. The next terms are the sum of two successive terms. In this article, generalizations of Fibonacci, Lucas and Mulatu (GFLM) numbers are built which are generalizations of the three types of numbers. The Binet formula is then built for the GFLM numbers, and determines the golden ratio, silver ratio and Bronze ratio of the GFLM numbers. This article also presents generalizations of these three types of ratios, called Metallic ratios. In the last part we state the Metallic ratio in the form of continued fraction and nested radicals.

Improved Least Common Multiple Effective Wavelength Method Used in Dual-Wavelength Interferometry

2021 ◽  
Vol 58 (7) ◽  
pp. 0712002
Author(s):  
郭小庭 Guo Xiaoting ◽  
刘晓军 Liu Xiaojun ◽  
雷自力 Lei Zili ◽  
杨文军 Yang Wenjun ◽  
徐龙 Xu Long
Keyword(s):  
Dual Wavelength ◽  
Effective Wavelength ◽  
Common Multiple ◽  
Least Common Multiple

The Grand Periodic Function

2018 ◽  
Author(s):  
Jan C. A. Boeyens
Keyword(s):  
Number Theory ◽  
Periodic Function ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Magic Number ◽  
Self Similarity ◽  
Covalent Interaction ◽  
Great Pyramid ◽  
Relationship Of ◽  
The Relationship

The discovery of material periodicity must rank as one of the major achievements of mankind. It reveals an ordered reality despite the gloomy pronouncements of quantum philosophers. Periodicity only appears in closed systems with well-defined boundary conditions. This condition excludes an infinite Euclidean universe and all forms of a chaotic multiverse. Manifestations of cosmic order were observed and misinterpreted by the ancients as divine regulation of terrestrial events, dictated by celestial intervention. Analysis of observed patterns developed into the ancient sciences of astrology, alchemy and numerology, which appeared to magically predict the effects of the macrocosm on the microcosm. The sciences of astronomy and chemistry have by now managed to outgrow the magic connotation, but number theory remains suspect as a scientific pursuit. The relationship between Fibonacci numbers and cosmic self-similarity is constantly being confused with spurious claims of religious and mystic codes, imagined to be revealed through the golden ratio in the architecture of the Great Pyramid and other structures such as the Temple of Luxor. The terminology which is shared by number theory and numerology, such as perfect number, magic number, tetrahedral number and many more, contributes to the confusion. It is not immediately obvious that number theory does not treat 3 as a sacred number, 13 as unlucky and 666 as an apocalyptic threat. The relationship of physical systems to numbers is no more mysterious nor less potent than to differential calculus. Like a differential equation, number theory does not dictate, but only describes physical behavior. The way in which number theory describes the periodicity of matter, atomic structure, superconductivity, electronegativity, bond order, and covalent interaction was summarized in a recent volume. The following brief summary of these results is augmented here by a discussion of atomic and molecular polarizabilities, as derived by number theory, and in all cases specified in relation to the grand periodic function that embodies self-similarity over all space-time.

On the least common multiple of Lucas subsequences

2013 ◽  
Vol 161 (4) ◽  
pp. 327-349 ◽  
Author(s):  
Shigeki Akiyama ◽  
Florian Luca
Keyword(s):  
Common Multiple ◽  
Least Common Multiple
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