scholarly journals On a problem of Kurt Mahler concerning binomial coefficents

1976 ◽  
Vol 14 (2) ◽  
pp. 299-302 ◽  
Author(s):  
Ian S. Williams
Keyword(s):  
Natural Number ◽  
Binomial Coefficients ◽  
Natural Numbers ◽  
Common Multiple ◽  
Least Common Multiple

Recently Kurt Mahler asked: for which natural numbers N is the least common multiple of all the binomial coefficients the product of the primes less than or equal to N? We obtain a formula for the least common multiple of all the binomial coefficients of any natural number N and hence show that 2,11, and 23 are the only soultions to Mahler's problem.

scholarly journals On the Least Common Multiple of Q-Binomial Coefficients

Integers ◽  
2010 ◽  
Vol 10 (3) ◽  
Author(s):  
Victor J. W. Guo
Keyword(s):  
Binomial Coefficients ◽  
Common Multiple ◽  
Least Common Multiple

AbstractWe first prove the following identitywhere

LH LABELING OF GRAPHS

2021 ◽  
Vol 10 (4) ◽  
pp. 2167-2179
Author(s):  
M. Farisa ◽  
K.S. Parvathy
Keyword(s):  
Common Factor ◽  
Natural Numbers ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Bijective Function

A graph $G$ with $n$ vertices is said to have an LH labeling if there exists a bijective function $f: V(G)$ to $\{1,2,3,\ldots ,n\} $ such that the induced map $f^*: E(G)\rightarrow N$, the set of natural numbers defined by $f^*(uv) = \frac{LCM(f(u), f(v))} {HCF(f(u),f(v))}$ is injective (LCM and HCF denotes the least common multiple and highest common factor respectively). A graph that admits an LH labeling is called an LH graph. This article explores the results of LH Labeling of some standard graphs.

scholarly journals A variant of the Hardy-Ramanujan theorem

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
M. Ram Murty ◽  
V Kumar Murty
Keyword(s):  
Natural Number ◽  
Positive Constant ◽  
Independent Interest ◽  
Prime Divisors ◽  
Normal Order ◽  
Common Multiple ◽  
Least Common Multiple

For each natural number $n$, we define $\omega^*(n)$ to be the number of primes $p$ such that $p-1$ divides $n$. We show that in contrast to the Hardy-Ramanujan theorem which asserts that the number $\omega(n)$ of prime divisors of $n$ has a normal order $\log\log n$, the function $\omega^*(n)$ does not have a normal order. We conjecture that for some positive constant $C$, $$\sum_{n\leq x} \omega^*(n)^2 \sim Cx(\log x). $$ Another conjecture related to this function emerges, which seems to be of independent interest. More precisely, we conjecture that for some constant $C>0$, as $x\to \infty$, $$\sum_{[p-1,q-1]\leq x} {1 \over [p-1, q-1]} \sim C \log x, $$ where the summation is over primes $p,q\leq x$ such that the least common multiple $[p-1,q-1]$ is less than or equal to $x$.

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

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 A unified treatment of syntax with binders

2012 ◽  
Vol 22 (4-5) ◽  
pp. 614-704 ◽  
Author(s):  
NICOLAS POUILLARD ◽  
FRANÇOIS POTTIER
Keyword(s):  
Natural Number ◽  
Data Structures ◽  
Transformation Function ◽  
Natural Numbers ◽  
Programming Error ◽  
Free Variables ◽  
De Bruijn ◽  
Abstract Interface ◽  
Representation Techniques ◽  
De Bruijn Indices

AbstractAtoms and de Bruijn indices are two well-known representation techniques for data structures that involve names and binders. However, using either technique, it is all too easy to make a programming error that causes one name to be used where another was intended. We propose an abstract interface to names and binders that rules out many of these errors. This interface is implemented as a library in Agda. It allows defining and manipulating term representations in nominal style and in de Bruijn style. The programmer is not forced to choose between these styles: on the contrary, the library allows using both styles in the same program, if desired. Whereas indexing the types of names and terms with a natural number is a well-known technique to better control the use of de Bruijn indices, we index types with worlds. Worlds are at the same time more precise and more abstract than natural numbers. Via logical relations and parametricity, we are able to demonstrate in what sense our library is safe, and to obtain theorems for free about world-polymorphic functions. For instance, we prove that a world-polymorphic term transformation function must commute with any renaming of the free variables. The proof is entirely carried out in Agda.

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
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