2,303 results on '"Prime factor"'
Search Results
2. On Cartesian products of signed graphs
- Author
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Dimitri Lajou, Laboratoire Bordelais de Recherche en Informatique (LaBRI), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB), and Lajou, Dimitri
- Subjects
Algebraic properties ,Of the form ,0102 computer and information sciences ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,01 natural sciences ,Combinatorics ,symbols.namesake ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,Prime factor ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Order (group theory) ,Chromatic scale ,0101 mathematics ,Signed graph ,Time complexity ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Cartesian product ,16. Peace & justice ,Graph ,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] ,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM] ,010201 computation theory & mathematics ,symbols ,Homomorphism ,Combinatorics (math.CO) ,Focus (optics) ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
In this paper, we study the Cartesian product of signed graphs as defined by Germina, Hameed and Zaslavsky (2011). Here we focus on its algebraic properties and look at the chromatic number of some Cartesian products. One of our main results is the unicity of the prime factor decomposition of signed graphs. This leads us to present an algorithm to compute this decomposition in linear time based on a decomposition algorithm for oriented graphs by Imrich and Peterin (2018). We also study the chromatic number of a signed graph, that is the minimum order of a signed graph to which the input signed graph admits a homomorphism, of graphs with underlying graph of the form P n □ P m , of Cartesian products of signed paths, of Cartesian products of signed complete graphs and of Cartesian products of signed cycles.
- Published
- 2022
3. Families of non-congruent numbers with odd prime factors of the form 8k + 3
- Author
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Wan Lee, Hayan Nam, Junguk Lee, and Myungjun Yu
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Combinatorics ,Algebra and Number Theory ,Prime factor ,Of the form ,Mathematics ,Congruent number - Published
- 2022
4. On the normal number of prime factors of sums of Fourier coefficients of eigenforms
- Author
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M. Ram Murty, Sudhir Pujahari, and V. Kumar Murty
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Combinatorics ,Riemann hypothesis ,symbols.namesake ,Algebra and Number Theory ,Log-log plot ,Prime factor ,symbols ,Normal number ,Fourier series ,Prime (order theory) ,Mathematics ,Normal order - Abstract
We study the normal number of prime factors of a f ( p ) + a g ( p ) with p prime and f , g distinct Hecke eigenforms of weight two. Assuming a quasi-generalized Riemann hypothesis, we show that the normal order is log log p . We also obtain an estimate for the number of primes p for which a f ( p ) + a g ( p ) = 0 .
- Published
- 2022
5. On the compactification of the Drinfeld modular curve of level Γ1Δ(n)
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Shin Hattori
- Subjects
Combinatorics ,Algebra and Number Theory ,010102 general mathematics ,Hodge bundle ,Modular form ,Prime factor ,010103 numerical & computational mathematics ,Compactification (mathematics) ,0101 mathematics ,01 natural sciences ,Modular curve ,Monic polynomial ,Mathematics - Abstract
Let p be a rational prime and q a power of p. Let n be a non-constant monic polynomial in F q [ t ] which has a prime factor of degree prime to q − 1 . In this paper, we define a Drinfeld modular curve Y 1 Δ ( n ) over A [ 1 / n ] and study the structure around cusps of its compactification X 1 Δ ( n ) , in a parallel way to Katz-Mazur's work on classical modular curves. Using them, we also define a Hodge bundle over X 1 Δ ( n ) such that Drinfeld modular forms of level Γ 1 ( n ) , weight k and some type are identified with global sections of its k-th tensor power.
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- 2022
6. New Semi-Prime Factorization and Application in Large RSA Key Attacks
- Author
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Anthony Overmars and Sitalakshmi Venkatraman
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Fermat's Last Theorem ,Discrete mathematics ,Lebesgue identity ,Brahmagupta–Fibonacci identity ,business.industry ,Mathematics::Number Theory ,Euler’s factorization ,Pythagorean theorem ,Cryptography ,Pythagorean triples ,semi-primes ,Factorization ,RSA cryptosystem ,Pythagorean triple ,Prime factor ,Pythagorean quadvery minimal, with most computations operating onruples ,Pythagorean quadruple ,T1-995 ,business ,Technology (General) ,Mathematics - Abstract
Semi-prime factorization is an increasingly important number theoretic problem, since it is computationally intractable. Further, this property has been applied in public-key cryptography, such as the Rivest–Shamir–Adleman (RSA) encryption systems for secure digital communications. Hence, alternate approaches to solve the semi-prime factorization problem are proposed. Recently, Pythagorean tuples to factor semi-primes have been explored to consider Fermat’s Christmas theorem, with the two squares having opposite parity. This paper is motivated by the property that the integer separating these two squares being odd reduces the search for semi-prime factorization by half. In this paper, we prove that if a Pythagorean quadruple is known and one of its squares represents a Pythagorean triple, then the semi-prime is factorized. The problem of semi-prime factorization is reduced to the problem of finding only one such sum of three squares to factorize a semi-prime. We modify the Lebesgue identity as the sum of four squares to obtain four sums of three squares. These are then expressed as four Pythagorean quadruples. The Brahmagupta–Fibonacci identity reduces these four Pythagorean quadruples to two Pythagorean triples. The greatest common divisors of the sides contained therein are the factors of the semi-prime. We then prove that to factor a semi-prime, it is sufficient that only one of these Pythagorean quadruples be known. We provide the algorithm of our proposed semi-prime factorization method, highlighting its complexity and comparative advantage of the solution space with Fermat’s method. Our algorithm has the advantage when the factors of a semi-prime are congruent to 1 modulus 4. Illustrations of our method for real-world applications, such as factorization of the 768-bit number RSA-768, are established. Further, the computational viabilities, despite the mathematical constraints and the unexplored properties, are suggested as opportunities for future research.
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- 2021
7. On a ternary Diophantine equation involving fractional powers with prime variables of a special form
- Author
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Fei Xue, Jinjiang Li, and Min Zhang
- Subjects
Combinatorics ,Algebra and Number Theory ,Number theory ,Diophantine equation ,Prime factor ,Multiplicity (mathematics) ,Ternary operation ,Prime (order theory) ,Real number ,Mathematics - Abstract
Let N be a sufficiently large real number. In this paper, it is proved that, for $$1
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- 2021
8. New Framework for Sequences With Perfect Autocorrelation and Optimal Crosscorrelation
- Author
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Min Kyu Song and Hong-Yeop Song
- Subjects
Combinatorics ,Set (abstract data type) ,Sequence ,Combinatorial design ,Integer ,Autocorrelation ,Prime factor ,Polyphase system ,Library and Information Sciences ,Quadrature amplitude modulation ,Computer Science Applications ,Information Systems ,Mathematics - Abstract
In this paper, we give a new framework for constructing perfect sequences, called generalized Milewski sequences, over various alphabets including Polyphase (PSK) as well as Amplitude-and-Polyphase (APSK) in general, and for constructing optimal sets of such perfect sequences by using combinatorial designs, called circular Florentine arrays. Specifically, we prove that, given any positive integer $m\geq 1$ , (i) there exists a perfect sequence of period $mN^{2}$ for any positive integer $N$ if there exists a perfect sequence (polyphase or not) of length $m$ ; (ii) an optimal $k$ -set of perfect sequences of length $mN^{2}$ can be constructed if there exist both a $k \times N$ circular Florentine array and an optimal $k$ -set of perfect sequences all of length $m$ . This enables us to find some optimal $k$ -set of perfect sequences where $k > p_{\text {min}}-1$ , where $p_{\text {min}}$ is the smallest prime factor of $mN^{2}$ .
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- 2021
9. On the number of semismooth integers
- Author
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Koji Suzuki
- Subjects
Combinatorics ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Algebra and Number Theory ,Computer Science::Information Retrieval ,Prime factor ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::General Literature ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,Analytic number theory ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
Let [Formula: see text] be a fixed positive number. Define [Formula: see text] as the number of positive integers [Formula: see text] having no prime factors [Formula: see text], and define [Formula: see text] as the number of positive integers [Formula: see text] having [Formula: see text] prime factors [Formula: see text], with all the other prime factors [Formula: see text]. In this paper, we give an asymptotic estimate for the ratio [Formula: see text], provided that [Formula: see text], [Formula: see text], and [Formula: see text] as [Formula: see text]. Also, combining this estimate with conventional ones for [Formula: see text], we provide sharp estimates for [Formula: see text].
- Published
- 2021
10. AN ELEMENTARY NOTE ON THE GREATEST PRIME FACTORS OF LINEARLY RELATED INTEGERS
- Author
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Mihai Caragiu
- Subjects
Combinatorics ,Algebra and Number Theory ,Prime factor ,Mathematics - Published
- 2021
11. Some remarks on small values of $$\tau (n)$$
- Author
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Anne Larsen and Kaya Lakein
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Conjecture ,Series (mathematics) ,Mathematics::Number Theory ,General Mathematics ,Function (mathematics) ,Congruence relation ,Ramanujan's sum ,Combinatorics ,symbols.namesake ,Integer ,Lucas number ,Prime factor ,symbols ,Mathematics - Abstract
A natural variant of Lehmer’s conjecture that the Ramanujan $$\tau $$ -function never vanishes asks whether, for any given integer $$\alpha $$ , there exist any $$n \in \mathbb {Z}^+$$ such that $$\tau (n) = \alpha $$ . A series of recent papers excludes many integers as possible values of the $$\tau $$ -function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for $$\tau (n)$$ . We synthesize these results and methods to prove that if $$0< \left| \alpha \right| < 100$$ and $$\alpha \notin T := \{2^k, -24,-48, -70,-90, 92, -96\}$$ , then $$\tau (n) \ne \alpha $$ for all $$n > 1$$ . Moreover, if $$\alpha \in T$$ and $$\tau (n) = \alpha $$ , then n is square-free with prescribed prime factorization. Finally, we show that a strong form of the Atkin-Serre conjecture implies that $$\left| \tau (n) \right| > 100$$ for all $$n > 2$$ .
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- 2021
12. Representasi verbal siswa sekolah menengah pertama dalam mengkomunikasikan faktor prima
- Author
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Fitry Wahyuni and Sukiyanto Sukiyanto
- Subjects
Prime factor ,ComputingMilieux_COMPUTERSANDEDUCATION ,Mathematics education ,Language of mathematics ,Subject (documents) ,Descriptive research ,Psychology ,Test (assessment) - Abstract
The purpose of this study was to describe students' communication in solving prime factors using written and oral communication. This study uses a qualitative and descriptive approach. The subjects of this study were 46 seventh grade students of Simanjaya Lamongan, East Java, Indonesia, Middle School who had obtained prime factor material. Data analysis was obtained by using a test of prime factors and interviews. Based on the results of data analysis, it was concluded that: 1) Written communication, as many as 40 students were able to communicate in writing with a percentage of 87% and 6 students were unable to communicate in writing with a percentage of 13%; 2) Oral communication, at this stage interviews were conducted by taking the subject as many as two students taken from one student who is able to communicate in writing and one student who is not able to communicate in writing, the results obtained that students are able to communicate in writing are also able to communicate verbally. verbally, while students who are not able to communicate in writing, it turns out that students are able to communicate orally. From the explanation above, it can be concluded that students are able to explain concepts into mathematical language, students are able to explain mathematical arithmetic operations, students are able to explain mathematical solutions and students are able to convey ideas or opinions. Thus, students can communicate in writing and orally well in solving prime factors.
- Published
- 2021
13. Modular Ternary Additive Problems with Irregular or Prime Numbers
- Author
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Olivier Ramaré and G. K. Viswanadham
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Von Mangoldt function ,Mathematics::Number Theory ,Liouville function ,010102 general mathematics ,Prime number ,Interval (mathematics) ,16. Peace & justice ,Möbius function ,01 natural sciences ,Prime (order theory) ,Ramanujan's sum ,Combinatorics ,symbols.namesake ,Mathematics (miscellaneous) ,0103 physical sciences ,Prime factor ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Our initial problem is to represent classes $$m$$ modulo $$q$$ by a sum of three terms, two being taken from rather small sets $$\mathcal A$$ and $$\mathcal B$$ and the third one having an odd number of prime factors (the so-called irregular numbers in S. Ramanujan’s terminology) and lying in an interval $$[q^{20r},q^{20r}+q^{16r}]$$ for some given $$r\ge1$$ . We show that it is always possible to do so provided that $$|\mathcal A|\cdot|\mathcal B|\ge q(\log q)^2$$ . The proof leads us to study the trigonometric polynomials over irregular numbers in a short interval and to seek very sharp bounds for them. We prove in particular that $$\sum_{q^{20r}\le s \le q^{20r}+q^{16r}}e(sa/q)\ll q^{16r}(\log q)/\sqrt{\varphi(q)}$$ uniformly in $$r$$ , where $$s$$ ranges over the irregular numbers. We develop a technique initiated by Selberg and Motohashi to do so. In short, we express the characteristic function of the irregular numbers via a family of bilinear decompositions akin to Iwaniec’s amplification process, and that uses pseudo-characters or local models. The technique applies to the Liouville function, to the Mobius function and also to the von Mangoldt function, in which case it is slightly more difficult. It is however simple enough to warrant explicit estimates, and we prove, for instance, that $$\bigl|\sum_{X
- Published
- 2021
14. Normal Sylow subgroups and monomial Brauer characters
- Author
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Xiaoyou Chen and Long Miao
- Subjects
Combinatorics ,Monomial ,Finite group ,Kernel (algebra) ,Mathematics (miscellaneous) ,Character (mathematics) ,Solvable group ,Sylow theorems ,Prime factor ,Mathematics - Abstract
Let G be a finite group, p be a prime divisor of |G|, and P be a Sylow p-subgroup of G. We prove that P is normal in a solvable group G if |G: ker φ|p′ = φ(1)p′ for every nonlinear irreducible monomial p-Brauer character φ of G, where ker φ is the kernel of φ and φ(1)p′ is the p′-part of φ(1).
- Published
- 2021
15. A ternary Diophantine inequality with prime numbers of a special form
- Author
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Jinjiang Li, Fei Xue, and Min Zhang
- Subjects
Combinatorics ,Arbitrarily large ,General Mathematics ,Prime factor ,Prime number ,Multiplicity (mathematics) ,Ternary operation ,Binary logarithm ,Prime (order theory) ,Mathematics ,Real number - Abstract
Let N be a sufficiently large real number. In this paper, we prove that, for $$10$$ , the Diophantine inequality $$\begin{aligned} \big |p_1^c+p_2^c+p_3^c-N\big
- Published
- 2021
16. An extension of the Siegel-Walfisz theorem
- Author
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Andreas Weingartner
- Subjects
Mathematics - Number Theory ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,Integer sequence ,Extension (predicate logic) ,Prime (order theory) ,Combinatorics ,Prime factor ,FOS: Mathematics ,11N25, 11N69 ,Number Theory (math.NT) ,Siegel–Walfisz theorem ,Mathematics - Abstract
We extend the Siegel-Walfisz theorem to a family of integer sequences that are characterized by constraints on the size of the prime factors. Besides prime powers, this family includes smooth numbers, almost primes and practical numbers., 11 pages
- Published
- 2021
17. A generalization of a theorem of Sylvester and Schur
- Author
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JiaLe Sheng and Hongguang Wu
- Subjects
Combinatorics ,Algebra and Number Theory ,Number theory ,Coprime integers ,Generalization ,Prime factor ,Least common multiple ,Mathematics - Abstract
In this paper, we give an estimate of the lower bounds of the least common multiple of $$a,a+b,\ldots ,a+kb$$ for $$(a,b)=1,k\in \textit{N}^+$$ . Precisely, we prove that for any two coprime positive integers a and b, we have $$\begin{aligned} L_{a,b,k}\ge \prod \limits _{p\mid b}p^{\text {Ord}_{p}^{k!}}\frac{1}{k!}\prod \limits _{i=0}^{k}(a+ib), \end{aligned}$$ where $$L_{a,b,k}$$ is the least common multiple of $$a,a+b,\ldots ,a+kb$$ and $$\text {Ord}_p^{k!}$$ denotes the least s for which $$p^s\mid k!$$ but $$p^{s+1}\not \mid k!$$ . In addition, we obtain a corollary that there is a number containing a prime divisor greater than k in the set $$\{a,a+b,\ldots ,a+kb\}$$ for $$(a,b)=1,b\ge 2$$ .
- Published
- 2021
18. Finite groups in which every self-centralizing subgroup is nilpotent or subnormal or a TI-subgroup
- Author
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Jiangtao Shi and Na Li
- Subjects
Mathematics::Group Theory ,Finite group ,Pure mathematics ,Nilpotent ,Ordinary differential equation ,Sylow theorems ,Prime factor ,Mathematics - Abstract
Let G be a finite group. We prove that if every self-centralizing subgroup of G is nilpotent or subnormal or a TI-subgroup, then every subgroup of G is nilpotent or subnormal. Moreover, G has either a normal Sylow p-subgroup or a normal p-complement for each prime divisor p of |G|.
- Published
- 2021
19. A Hardy–Ramanujan-type inequality for shifted primes and sifted sets
- Author
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Kevin Ford
- Subjects
Inequality ,Mathematics::Number Theory ,General Mathematics ,media_common.quotation_subject ,Type inequality ,Ramanujan's sum ,Combinatorics ,symbols.namesake ,Number theory ,Ordinary differential equation ,Prime factor ,symbols ,Mathematics ,media_common - Abstract
We establish an analog of the Hardy–Ramanujan inequality for counting members of sifted sets with a given number of distinct prime factors. In particular, we establish a bound for the number of shifted primes p + a below x with k distinct prime factors, uniformly for all positive integers k.
- Published
- 2021
20. Factorization of Dickson polynomials over finite fields
- Author
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Fabio Enrique Brochero Martínez and Nelcy Esperanza Arévalo Baquero
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Pure mathematics ,General Mathematics ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Finite field ,Computational Theory and Mathematics ,Factorization ,010201 computation theory & mathematics ,Prime factor ,0202 electrical engineering, electronic engineering, information engineering ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Let $$D_n(x;a)$$ and $$E_n(x;a)\in {\mathbb {F}}_q[x]$$ be Dickson polynomials of first and second kind respectively, where $${\mathbb {F}}_q$$ is a finite field with q elements. In this article we show explicitly the irreducible factors of these polynomials in the case that every prime divisor of n divides $$q-1$$ . This result generalizes the results found in (Finite Fields Appl. 3:84–96, 1997; Explicit factorization of cyclotomic and Dickson polynomials over finite fields, Springer, Berlin, 2007; Finite Fields Appl. 38:40-56, 2016; Discrete Math. 342:111618, 2019).
- Published
- 2021
21. Three conjectures on P+(n) and P+(n + 1) hold under the Elliott-Halberstam conjecture for friable integers
- Author
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Zhiwei Wang
- Subjects
Algebra and Number Theory ,Conjecture ,Elliott–Halberstam conjecture ,010102 general mathematics ,Integer sequence ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Integer ,Prime factor ,Natural density ,0101 mathematics ,Dickman function ,Mathematics - Abstract
Denote by P + ( n ) the largest prime factor of an integer n. In this paper, we show that the Elliott-Halberstam conjecture for friable integers (or smooth integers) implies three conjectures concerning the largest prime factors of consecutive integers, formulated by Erdős-Turan in the 1930s, by Erdős-Pomerance in 1978, and by Erdős in 1979 respectively. More precisely, assuming the Elliott-Halberstam conjecture for friable integers, we deduce that the three sets E 1 = { n ⩽ x : P + ( n ) ⩽ x s , P + ( n + 1 ) ⩽ x t } , E 2 = { n ⩽ x : P + ( n ) P + ( n + 1 ) x α } , E 3 = { n ⩽ x : P + ( n ) P + ( n + 1 ) } have an asymptotic density ρ ( 1 / s ) ρ ( 1 / t ) , ∫ T α u ( y ) u ( z ) d y d z , 1/2 respectively for s , t ∈ ( 0 , 1 ) , where ρ ( ⋅ ) is the Dickman function, and T α , u ( ⋅ ) are defined in Theorem 2.
- Published
- 2021
22. On key applications of the Turán–Kubilius inequality
- Author
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Jean-Marie De Koninck and Imre Kátai
- Subjects
Discrete mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Natural number ,01 natural sciences ,Set (abstract data type) ,010104 statistics & probability ,Number theory ,Law of large numbers ,Ordinary differential equation ,Prime factor ,Key (cryptography) ,0101 mathematics ,media_common ,Mathematics - Abstract
After briefly describing the origin and the scope of the Turan–Kubilius inequality, we show how this important inequality leads to the law of large numbers for the prime factors belonging to the middle region of a random natural number. Finally, we show how the same idea is implemented for a sparser subset, the set of shifted primes.
- Published
- 2021
23. Iterative Power Algorithm for Global Optimization with Quantics Tensor Trains
- Author
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Micheline B. Soley, Paul Bergold, and Victor S. Batista
- Subjects
Quantum Physics ,Optimization problem ,010304 chemical physics ,FOS: Physical sciences ,Dirac delta function ,01 natural sciences ,Computer Science Applications ,symbols.namesake ,Power iteration ,0103 physical sciences ,Prime factor ,symbols ,Probability distribution ,Tensor ,Physical and Theoretical Chemistry ,Quantum Physics (quant-ph) ,Global optimization ,Algorithm ,Curse of dimensionality - Abstract
Optimization algorithms play a central role in chemistry since optimization is the computational keystone of most molecular and electronic structure calculations. Herein, we introduce the iterative power algorithm (IPA) for global optimization and a formal proof of convergence for both discrete and continuous global search problems, which is essential for applications in chemistry such as molecular geometry optimization. IPA implements the power iteration method in quantics tensor train (QTT) representations. Analogous to the imaginary time propagation method with infinite mass, IPA starts with an initial probability distribution ρ0(x) and iteratively applies the recurrence relation ρk+1(x) = U(x) ρk(x)/∥Uρk∥L1, where U(x) = e-V(x) is defined in terms of the potential energy surface (PES) V(x) with global minimum at x = x*. Upon convergence, the probability distribution becomes a delta function δ(x - x*), so the global minimum can be obtained as the position expectation value x* = Tr[x δ(x - x*)]. QTT representations of V(x) and ρ(x) are generated by fast adaptive interpolation of multidimensional arrays to bypass the curse of dimensionality and the need to evaluate V(x) for all possible values of x. We illustrate the capabilities of IPA for global search optimization of two multidimensional PESs, including a differentiable model PES of a DNA chain with D = 50 adenine-thymine base pairs, and a discrete non-differentiable potential energy surface, V(p) = mod(N,p), that resolves the prime factors of an integer N, with p in the space of prime numbers {2, 3,..., pmax} folded as a d-dimensional 21 × 22 × ··· × 2d tensor. We find that IPA resolves multiple degenerate global minima even when separated by large energy barriers in the highly rugged landscape of the potentials. Therefore, IPA should be of great interest for a wide range of other optimization problems ubiquitous in molecular and electronic structure calculations.
- Published
- 2021
24. Counting multiplicative groups with prescribed subgroups
- Author
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Jenna Downey and Greg Martin
- Subjects
Surface (mathematics) ,Discrete mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,Multiplicative group ,Computer Science::Information Retrieval ,Multiplicative function ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Number theory ,Counting problem ,Prime factor ,FOS: Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Computer Science::General Literature ,Number Theory (math.NT) ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors. First, given an odd prime $q$ and a finite abelian $q$-group $H$, we consider the set of integers $n\le x$ such that the Sylow $q$-subgroup of the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$ is isomorphic to $H$. We show that the counting function of this set of integers is asymptotic to $K x(\log\log x)^\ell/(\log x)^{1/(q-1)}$ for explicit constants $K$ and $\ell$ depending on $q$ and $H$. Second, we consider the set of integers $n\le x$ such that the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$ is "maximally non-cyclic", that is, such that all of its prime-power subgroups are elementary groups. We show that the counting function of this set of integers is asymptotic to $A x/(\log x)^{1-\xi}$ for an explicit constant $A$, where $\xi$ is Artin's constant. As it turns out, both of these group-theoretic problems can be reduced to problems of counting integers with restrictions on their prime factors, allowing them to be addressed by classical techniques of analytic number theory., Comment: 21 pages
- Published
- 2021
25. On computing the degree of a Chebyshev Polynomial from its value
- Author
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Erdal Imamoglu and Erich Kaltofen
- Subjects
Chebyshev polynomials ,Chebyshev Polynomials ,Algebra and Number Theory ,Logarithm ,Modulo ,Prime number ,Chebyshev filter ,Polynomials of the First Kind ,Combinatorics ,Computational Mathematics ,Factorization ,Prime factor ,Discrete logarithms ,Linear combination ,Algorithms ,Interpolation in terms of the Chebyshev ,Mathematics - Abstract
Algorithms for interpolating a polynomial f from its evaluation points whose running time depends on the sparsity to f the polynomial when it is represented as a linear combination of t Chebyshev Polynomials of the First Kind with non-zero scalar coefficients are given by Lakshman and Saunders (1995), Kaltofen and Lee (2003) and Arnold and Kaltofen (2015). The term degrees are computed from values of Chebyshev Polynomials of those degrees. We give an algorithm that computes those degrees in the manner of the Pohlig and Hellman algorithm (1978) for computing discrete logarithms modulo a prime number p when the factorization of p - 1(or p + 1) has small prime factors, that is, when p - 1(or p + 1) is smooth. Our algorithm can determine the Chebyshev degrees modulo such primes in bit complexity log(p)(O(1)) times the squareroot of the largest prime factor of p - 1( or p + 1). (C) 2020 Elsevier Ltd. All rights reserved. National Science FoundationNational Science Foundation (NSF) [CCF-1421128, CCF-1708884] Supported by National Science Foundation CCF-1421128 and CCF-1708884.
- Published
- 2021
26. An additive problem over Piatetski–Shapiro primes and almost-primes
- Author
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Min Zhang, Fei Xue, and Jinjiang Li
- Subjects
Combinatorics ,Algebra and Number Theory ,Number theory ,010201 computation theory & mathematics ,010102 general mathematics ,Prime factor ,Multiplicity (mathematics) ,0102 computer and information sciences ,0101 mathematics ,Type (model theory) ,01 natural sciences ,Mathematics - Abstract
Let $$\mathcal {P}_r$$ denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we establish a theorem of Bombieri–Vinogradov type for the Piatetski–Shapiro primes $$p=[n^{1/\gamma }]$$ with $$\frac{85}{86}
- Published
- 2021
27. On additive and multiplicative decompositions of sets of integers with restricted prime factors, I. (Smooth numbers)
- Author
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Kálmán Győry, Lajos Hajdu, and András Sárközy
- Subjects
Sequence ,Conjecture ,Mathematics - Number Theory ,General Mathematics ,Sieve (category theory) ,010102 general mathematics ,Multiplicative function ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Prime factor ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Unit (ring theory) ,Mathematics - Abstract
In Sarkozy (2001) the third author of this paper presented two conjectures on the additive decomposability of the sequence of ”smooth” (or ”friable”) numbers. Elsholtz and Harper (2015) proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of S -unit equations).
- Published
- 2021
28. Prime Factors and Divisibility of Sums of Powers of Fibonacci and Lucas Numbers
- Author
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Mariah Michael and Spirit Karcher
- Subjects
Combinatorics ,Fibonacci number ,Sums of powers ,Lucas number ,Prime factor ,General Medicine ,Divisibility rule ,Mathematics - Abstract
The Fibonacci sequence, whose first terms are f0; 1; 1; 2; 3; 5; : : :g, is generated using the recursive formula Fn+2 = Fn+1 + Fn with F0 = 0 and F1 = 1. This sequence is one of the most famous integer sequences because of its fascinating mathematical properties and connections with other fields such as biology, art, and music. Closely related to the Fibonacci sequence is the Lucas sequence. The Lucas sequence, whose first terms are f2; 1; 3; 4; 7; 11; : : :g, is generated using the recursive formula Ln+2 = Ln+1 + Ln with L0 = 2 and L1 = 1. In this paper, patterns in the prime factors of sums of powers of Fibonacci and Lucas numbers are examined. For example, F2 3n+4 + F2 3n+2 is even for all n 2 N0. To prove these results, techniques from modular arithmetic and facts about the divisibility of Fibonacci and Lucas numbers are utilized. KEYWORDS: Fibonacci Sequence; Lucas Sequence; Modular Arithmetic; Divisibility Sequence
- Published
- 2021
29. A conjecture of Watkins for quadratic twists
- Author
-
Jose A. Esparza-Lozano and Hector Pasten
- Subjects
Combinatorics ,Elliptic curve ,Conjecture ,Quadratic equation ,Rank (linear algebra) ,Degree (graph theory) ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,Prime factor ,Square-free integer ,Mathematics - Abstract
Watkins conjectured that for an elliptic curve $E$ over $\mathbb{Q}$ of Mordell-Weil rank $r$, the modular degree of $E$ is divisible by $2^r$. If $E$ has non-trivial rational $2$-torsion, we prove the conjecture for all the quadratic twists of $E$ by squarefree integers with sufficiently many prime factors.
- Published
- 2021
30. On sum of prime factors of composite positive integers
- Author
-
Yuchen Ding and Xiaodong Lü
- Subjects
Algebra and Number Theory ,Conjecture ,010102 general mathematics ,Composite number ,Prime number ,0102 computer and information sciences ,01 natural sciences ,Upper and lower bounds ,Combinatorics ,Number theory ,010201 computation theory & mathematics ,Prime factor ,0101 mathematics ,Mathematics - Abstract
Let $${\mathfrak {B}}(x)$$ be the number of composite positive integers up to x whose sum of distinct prime factors is a prime number. Luca and Moodley proved that there exist two positive constants $$a_1$$ and $$a_2$$ such that $$\begin{aligned} a_1x/\log ^3x\le {\mathfrak {B}}(x)\le a_2x/\log x. \end{aligned}$$ Assuming a uniform version of the Bateman–Horn conjecture, they gave a conditional proof of a lower bound of the same order of magnitude as the upper bound. In this paper, we offer an unconditional proof of the this result, i.e., $$\begin{aligned} {\mathfrak {B}}(x)\asymp \frac{x}{\log x}. \end{aligned}$$
- Published
- 2021
31. On small sets of integers
- Author
-
Salvatore Tringali and Paolo Leonetti
- Subjects
Upper and lower densities ,Ideals on sets ,Large and small sets (of integers) ,Star (game theory) ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,Primary 11B05, 28A10, Secondary 39B62 ,Prime factor ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Natural density ,Number Theory (math.NT) ,0101 mathematics ,Mathematics ,Polynomial (hyperelastic model) ,Algebra and Number Theory ,Mathematics - Number Theory ,Zero set ,Image (category theory) ,010102 general mathematics ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Number theory ,Mathematics - Classical Analysis and ODEs ,010201 computation theory & mathematics ,Binary quadratic form - Abstract
An upper quasi-density on $\bf H$ (the integers or the non-negative integers) is a real-valued subadditive function $\mu^\ast$ defined on the whole power set of $\mathbf H$ such that $\mu^\ast(X) \le \mu^\ast({\bf H}) = 1$ and $\mu^\ast(k \cdot X + h) = \frac{1}{k}\, \mu^\ast(X)$ for all $X \subseteq \bf H$, $k \in {\bf N}^+$, and $h \in \bf N$, where $k \cdot X := \{kx: x \in X\}$; and an upper density on $\bf H$ is an upper quasi-density on $\bf H$ that is non-decreasing with respect to inclusion. We say that a set $X \subseteq \bf H$ is small if $\mu^\ast(X) = 0$ for every upper quasi-density $\mu^\ast$ on $\bf H$. Main examples of upper densities are given by the upper analytic, upper Banach, upper Buck, and upper P\'olya densities, along with the uncountable family of upper $\alpha$-densities, where $\alpha$ is a real parameter $\ge -1$ (most notably, $\alpha = -1$ corresponds to the upper logarithmic density, and $\alpha = 0$ to the upper asymptotic density). It turns out that a subset of $\bf H$ is small if and only if it belongs to the zero set of the upper Buck density on $\bf Z$. This allows us to show that many interesting sets are small, including the integers with less than a fixed number of prime factors, counted with multiplicity; the numbers represented by a binary quadratic form with integer coefficients whose discriminant is not a perfect square; and the image of $\bf Z$ through a non-linear integral polynomial in one variable., Comment: 15 pp, no figures. The paper is a sequel of arXiv:1506.04664. Fixed minor details. To appear in The Ramanujan Journal
- Published
- 2021
32. On the Waring–Goldbach problem for squares, cubes and higher powers
- Author
-
Min Zhang and Jinjiang Li
- Subjects
Algebra and Number Theory ,010102 general mathematics ,Multiplicity (mathematics) ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,symbols.namesake ,Number theory ,Integer ,010201 computation theory & mathematics ,Fourier analysis ,Waring–Goldbach problem ,Prime factor ,symbols ,Exponent ,0101 mathematics ,Ternary operation ,Mathematics - Abstract
Let $$\mathcal {P}_r$$ denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we generalize the result of Vaughan (Bull Lond Math Soc 17(1):17–20, 1985) for ternary ‘admissible exponent’. Moreover, we use the refined ‘admissible exponent’ to prove that, for $$3\leqslant k\leqslant 14$$ and for every sufficiently large even integer n, the following equation $$\begin{aligned} n=x^2+p_1^2+p_2^3+p_3^3+p_4^3+p_5^k \end{aligned}$$ is solvable with x being an almost-prime $$\mathcal {P}_{r(k)}$$ and the other variables primes, where r(k) is defined in Theorem 1.1. This result constitutes a deepening of previous results.
- Published
- 2021
33. Oscillations in weighted arithmetic sums
- Author
-
Tim Trudgian and Michael J. Mossinghoff
- Subjects
Pure mathematics ,Algebra and Number Theory ,Liouville function ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Riemann hypothesis ,symbols.namesake ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,010201 computation theory & mathematics ,Prime factor ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,symbols ,Arithmetic function ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
We examine oscillations in a number of sums of arithmetic functions involving [Formula: see text], the total number of prime factors of [Formula: see text], and [Formula: see text], the number of distinct prime factors of [Formula: see text]. In particular, we examine oscillations in [Formula: see text] and in [Formula: see text] for [Formula: see text], and in [Formula: see text]. We show for example that each of the inequalities [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] is true infinitely often, disproving some hypotheses of Sun.
- Published
- 2021
34. On the Erdős primitive set conjecture in function fields
- Author
-
Charlotte Kavaler, Andrés Gómez-Colunga, Nathan McNew, and Mirilla Zhu
- Subjects
Combinatorics ,Algebra and Number Theory ,Conjecture ,010102 general mathematics ,Prime factor ,Multiplicity (mathematics) ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Monic polynomial ,Function field ,Mathematics - Abstract
Erdős proved that F ( A ) : = ∑ a ∈ A 1 a log a converges for any primitive set of integers A and later conjectured this sum is maximized when A is the set of primes. Banks and Martin further conjectured that F ( P 1 ) > ⋯ > F ( P k ) > F ( P k + 1 ) > ⋯ , where P j is the set of integers with j prime factors counting multiplicity, though this was recently disproven by Lichtman. We consider the corresponding problems over the function field F q [ x ] , investigating the sum F ( A ) : = ∑ f ∈ A 1 deg f ⋅ q deg f . We establish a uniform bound for F ( A ) over all primitive sets of polynomials A ⊂ F q [ x ] and conjecture that it is maximized by the set of monic irreducible polynomials. We find that the analogue of the Banks-Martin conjecture is false for q = 2 , 3, and 4, but we find computational evidence that it holds for q > 4 .
- Published
- 2021
35. On some topological and combinatorial lower bounds on the chromatic number of Kneser type hypergraphs
- Author
-
Amir Jafari and Soheil Azarpendar
- Subjects
Combinatorics ,Hypergraph ,Conjecture ,Computational Theory and Mathematics ,Generalization ,Prime factor ,Discrete Mathematics and Combinatorics ,Chromatic scale ,Type (model theory) ,Upper and lower bounds ,Theoretical Computer Science ,Vertex (geometry) ,Mathematics - Abstract
In this paper, we prove a generalization of a conjecture of Erdos, about the chromatic number of certain Kneser-type hypergraphs. For integers n , k , r , s with n ≥ r k and 2 ≤ s ≤ r , the r-uniform general Kneser hypergraph KG s r ( n , k ) , has all k-subsets of { 1 , … , n } as the vertex set and all multi-sets { A 1 , … , A r } of k-subsets with s-wise empty intersections as the edge set. The case r = s = 2 , was considered by Kneser [7] in 1955, where he conjectured that its chromatic number is n − 2 ( k − 1 ) . This was finally proved by Lovasz [9] in 1978. The case r > 2 and s = 2 , was considered by Erdos in 1973, and he conjectured that its chromatic number is ⌈ n − r ( k − 1 ) r − 1 ⌉ . This conjecture was proved by Alon, Frankl and Lovasz [2] in 1986. The case where s > 2 , was considered by Sarkaria [11] in 1990, where he claimed to prove a lower bound for its chromatic number which generalized all previous results. Unfortunately, an error was found by Lange and Ziegler [14] in 2006 in the induction method of Sarkaria on the number of prime factors of r, and Sarkaria's proof only worked when s is less than the smallest prime factor of r or s = 2 . Finally in 2019, Aslam, Chen, Coldren, Frick and Setiabrata [3] were able to extend this by using methods different from Sarkaria to the case when r = 2 α 0 p 1 α 1 … p t α t and 2 ≤ s ≤ 2 α 0 ( p 1 − 1 ) α 1 … ( p t − 1 ) α t . In this paper, by applying the Z p -Tucker lemma of Ziegler [13] and Meunier [10] , we finally prove the general Erdos conjecture and prove the claimed result of Sarkaria for any 2 ≤ s ≤ r . We also provide another proof of a special case of this result, using methods similar to those of Alon, Frankl, and Lovasz [2] and compute the connectivity of certain simplicial complexes that might be of interest in their own right.
- Published
- 2021
36. IND-CCA1 Secure FHE on Non-Associative Ring
- Author
-
Masahiro Yagisawa
- Subjects
Ring (mathematics) ,Theoretical computer science ,Computer science ,business.industry ,Applied Mathematics ,Signal Processing ,Prime factor ,Bootstrapping (linguistics) ,Cloud computing ,Electrical and Electronic Engineering ,business ,Computer Graphics and Computer-Aided Design ,Associative property - Published
- 2021
37. On the exceptional set of transcendental functions with integer coefficients in a prescribed set: The Problems A and C of Mahler
- Author
-
Carlos Gustavo Moreira and Diego Marques
- Subjects
Discrete mathematics ,Algebra and Number Theory ,Complex conjugate ,Transcendental function ,010102 general mathematics ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Set (abstract data type) ,Integer ,Bounded function ,Prime factor ,0101 mathematics ,Element (category theory) ,Mathematics - Abstract
In 1976, Mahler posed the question about the existence of a transcendental function f ∈ Z { z } with bounded coefficients and such that f ( Q ‾ ∩ B ( 0 , 1 ) ) ⊆ Q ‾ . In this paper, we prove, in particular, the existence of such a function but with the weaker requirement that the coefficients have only 2 and 3 as prime factors. More generally, we shall prove that any subset of Q ‾ ∩ B ( 0 , 1 ) , which is closed under complex conjugation and which contains the element 0, is the exceptional set of uncountably many transcendental functions in Z { z } with coefficients having only 2 and 3 as prime factors.
- Published
- 2021
38. Some properties of Zumkeller numbers and k-layered numbers
- Author
-
Daniel Yaqubi, Pankaj Jyoti Mahanta, and Manjil P. Saikia
- Subjects
Algebra and Number Theory ,Mathematics - Number Theory ,Mathematics::Number Theory ,Harmonic mean ,010102 general mathematics ,Sigma ,010103 numerical & computational mathematics ,11A25, 11B75, 11D99 ,01 natural sciences ,Combinatorics ,Integer ,Prime factor ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,QA ,Perfect number ,Mathematics - Abstract
Generalizing the concept of a perfect number is a Zumkeller or integer perfect number that was introduced by Zumkeller in 2003. The positive integer $n$ is a Zumkeller number if its divisors can be partitioned into two sets with the same sum, which will be $\sigma(n)/2$. Generalizing even further, we call $n$ a $k$-layered number if its divisors can be partitioned into $k$ sets with equal sum. In this paper, we completely characterize Zumkeller numbers with two distinct prime factors and give some bounds for prime factorization in case of Zumkeller numbers with more than two distinct prime factors. We also characterize $k$-layered numbers with two distinct prime factors and even $k$-layered numbers with more than two distinct odd prime factors. Some other results concerning these numbers and their relationship with practical numbers and Harmonic mean numbers are also discussed., Comment: 14 pages, accepted version, to appear in the Journal of Number Theory
- Published
- 2020
39. 2-Adic properties for the numbers of representations in the alternating groups
- Author
-
Yugen Takegahara
- Subjects
Physics ,010505 oceanography ,General Mathematics ,010102 general mathematics ,Alternating group ,Order (ring theory) ,Cyclic group ,01 natural sciences ,Combinatorics ,Integer ,Symmetric group ,Prime factor ,Exponent ,0101 mathematics ,Direct product ,0105 earth and related environmental sciences - Abstract
Let A be the direct product of a cyclic group of order $$2^u$$ with $$u\ge 1$$ and a cyclic group of order $$2^v$$ with $$u\ge v\ge 0$$ . There are some 2-adic properties of the number $$h(A,A_n)$$ of homomorphisms from A to the alternating group $$A_n$$ on n-letters, which are similar to those of the number of homomorphisms from A to the symmetric group on n-letters. The exponent of 2 in the decomposition of $$h(A,A_n)$$ into prime factors is denoted by $${\mathrm {ord}}_2(h(A,A_n))$$ . Let [x] denote the largest integer not exceeding a real number x. For any nonnegative integer n, the lower bound of $${\mathrm {ord}}_2(h(A,A_n))$$ is $$\sum _{j=1}^u[n/2^j]+[n/2^{u+2}]-[n/2^{u+3}]-1$$ if $$u=v\ge 1$$ , and is $$\sum _{j=1}^u[n/2^j]-(u-v)[n/2^{u+1}]-1$$ otherwise. For any positive odd integer y, $${\mathrm {ord}}_2(h(A,A_{2^{u+1}y}))$$ and $${\mathrm {ord}}_2(h(A,A_{2^{u+1}y+1}))$$ are described by certain 2-adic integers if either $$u\ge v+2\ge 3$$ or $$u\ge 1$$ and $$v=0$$ . The values $$\{h(A,A_n)\}_{n=0}^\infty $$ are explained by certain 2-adic analytic functions unless $$u=v+1\ge 2$$ . The results are obtained by using the generating function $$\sum _{n=0}^\infty h(A,A_n)X^n/n!$$ .
- Published
- 2020
40. Free cyclic group actions on highly-connected 2n-manifolds
- Author
-
Jianqiang Yang and Yang Su
- Subjects
Class (set theory) ,Pure mathematics ,57R65, 57R19, 57S17, 57S25 ,Applied Mathematics ,General Mathematics ,Homotopy ,010102 general mathematics ,Geometric Topology (math.GT) ,Cyclic group ,01 natural sciences ,Mathematics - Geometric Topology ,0103 physical sciences ,Prime factor ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Constant (mathematics) ,Topological conjugacy ,Mathematics - Abstract
In this paper we study smooth orientation-preserving free actions of the cyclic group $\mathbb Z/m$ on a class of $(n-1)$-connected $2n$-manifolds, $\sharp g (S^n \times S^n)\sharp \Sigma$, where $\Sigma$ is a homotopy $2n$-sphere. When $n=2$ we obtain a classification up to topological conjugation. When $n=3$ we obtain a classification up to smooth conjugation. When $n \ge 4$ we obtain a classification up to smooth conjugation when the prime factors of $m$ are larger than a constant $C(n)$., Comment: 18 pages
- Published
- 2020
41. PENINGKATAN PEMAHAMAN KONSEP FAKTORISASI PRIMA DAN APLIKASINYA MENGGUNAKAN MEDIA KREATIVITAS SISWA BAGI GURU SD NEGERI ENTROP JAYAPURA PROVINSI PAPUA
- Author
-
Westy B. Kawuwung, Bonefasius Yanwar Boy, and Epiphani I. Y. Palit
- Subjects
Service (business) ,School teachers ,Mathematical problem ,Prime factor ,Mathematics education - Abstract
The aim of this service activity is to increase the understanding of Entrop Elementary School teachers about one of the basic concepts of mathematics, namely prime factorization, its application in solving mathematical problems, as well as creative aids that can be used in teaching the concept to students. The method used is the explanation of mathematical concepts about factors, prime factors, and prime factorization of a number with examples of its application in solving mathematical problems. After that, examples of teaching aids which can be used to explain prime factorization to students are given. The result of this activity is an increase in the understanding of Entrop Public Elementary School teachers about the mathematical concepts of prime factorization, this can be seen by comparing their pre-test and post-test scores. Of the 23 teachers who participated in the activity, 80% experienced an increase in the understanding of the concept.Keywords: entrop, factors, prime factors, prime factorization, mathematic
- Published
- 2020
42. On a sum involving the number of distinct prime factors function related to the integer part function
- Author
-
Mihoub Bouderbala and Meselem Karras
- Subjects
Combinatorics ,Mathematics::Combinatorics ,Computer Science::Discrete Mathematics ,Prime factor ,Function (mathematics) ,Computer Science::Computational Geometry ,Mathematics ,Integer (computer science) - Abstract
In this paper, we obtain asymptotic formula on the sum \sum\limits_{n\leq x}\omega \left( \left\lfloor \frac{x}{n}\right\rfloor \right) , where \omega \left( n\right) denote the number of distinct prime divisors of n and \left\lfloor t\right\rfloor denotes the integer part of t.
- Published
- 2020
43. On the prime factors of the iterates of the Ramanujan τ–function
- Author
-
Pantelimon Stanica, Florian Luca, and Sibusiso Mabaso
- Subjects
General Mathematics ,Diophantine equation ,010102 general mathematics ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Ramanujan's sum ,Combinatorics ,symbols.namesake ,Integer ,Iterated function ,Prime factor ,symbols ,0101 mathematics ,Mathematics - Abstract
In this paper, for a positive integer n ≥ 1, we look at the size and prime factors of the iterates of the Ramanujan τ function applied to n.
- Published
- 2020
44. Statistical independence in mathematics–the key to a Gaussian law
- Author
-
Gunther Leobacher and Joscha Prochno
- Subjects
General Mathematics ,010102 general mathematics ,Probability axioms ,Context (language use) ,Divisibility rule ,01 natural sciences ,010104 statistics & probability ,Number theory ,Law ,Prime factor ,Limit (mathematics) ,0101 mathematics ,Lacunary function ,Central limit theorem ,Mathematics - Abstract
In this manuscript we discuss the notion of (statistical) independence embedded in its historical context. We focus in particular on its appearance and role in number theory, concomitantly exploring the intimate connection of independence and the famous Gaussian law of errors. As we shall see, this at times requires us to go adrift from the celebrated Kolmogorov axioms, which give the appearance of being ultimate ever since they have been introduced in the 1930s. While these insights are known to many a mathematician, we feel it is time for both a reminder and renewed awareness. Among other things, we present the independence of the coefficients in a binary expansion together with a central limit theorem for the sum-of-digits function as well as the independence of divisibility by primes and the resulting, famous central limit theorem of Paul Erdős and Mark Kac on the number of different prime factors of a number $$n\in{\mathbb{N}}$$ n ∈ N . We shall also present some of the (modern) developments in the framework of lacunary series that have its origin in a work of Raphaël Salem and Antoni Zygmund.
- Published
- 2020
45. Domination in direct products of complete graphs
- Author
-
Harish Vemuri
- Subjects
Cayley graph ,Domination analysis ,Applied Mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,0102 computer and information sciences ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,Unitary state ,Interpretation (model theory) ,Combinatorics ,05C69 ,010201 computation theory & mathematics ,Prime factor ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Arithmetic function ,Combinatorics (math.CO) ,Mathematics - Abstract
Let $X_{n}$ denote the unitary Cayley graph of $\mathbb{Z}/n\mathbb{Z}$. We continue the study of cases in which the inequality $\gamma_t(X_n) \le g(n)$ is strict, where $\gamma_t$ denotes the total domination number, and $g$ is the arithmetic function known as Jacobsthal's function. The best that is currently known in this direction is a construction of Burcroff which gives a family of $n$ with arbitrarily many prime factors that satisfy $\gamma_t(X_n) \le g(n)-2$. We present a new interpretation of the problem which allows us to use recent results on the computation of Jacobsthal's function to construct $n$ with arbitrarily many prime factors that satisfy $\gamma_t(X_n) \le g(n)-16$. We also present new lower bounds on the domination numbers of direct products of complete graphs, which in turn allow us to derive new asymptotic lower bounds on $\gamma(X_n)$, where $\gamma$ denotes the domination number. Finally, resolving a question of Defant and Iyer, we completely classify all graphs $G = \prod_{i=1}^t K_{n_i}$ satisfying $\gamma(G) = t+2$., Comment: 15 pages
- Published
- 2020
46. On finiteness of odd superperfect numbers
- Author
-
Tomohiro Yamada
- Subjects
010101 applied mathematics ,Combinatorics ,Mathematics - Number Theory ,Prime factor ,FOS: Mathematics ,Mathematics::Metric Geometry ,Number Theory (math.NT) ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Primary 11A25, Secondary 11A05, 11D61, 11J86 ,Mathematics - Abstract
Some new results concerning the equation $\sigma(N)=aM, \sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors., Comment: 14 pages, the author's final version, with the link to Western Number Theory Problems updated
- Published
- 2020
47. THE CASIMIR NUMBER AND THE DETERMINANT OF A FUSION CATEGORY
- Author
-
Zhihua Wang, Libin Li, and Gongxiang Liu
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Field (mathematics) ,01 natural sciences ,Casimir effect ,03 medical and health sciences ,0302 clinical medicine ,Prime factor ,Exponent ,030212 general & internal medicine ,0101 mathematics ,Cauchy's integral theorem ,Algebraically closed field ,Complex number ,Mathematics - Abstract
Let $\mathcal{C}$ be a fusion category over an algebraically closed field $\mathbb{k}$ of arbitrary characteristic. Two numerical invariants of $\mathcal{C}$ , that is, the Casimir number and the determinant of $\mathcal{C}$ are considered in this paper. These two numbers are both positive integers and admit the property that the Grothendieck algebra $(\mathcal{C})\otimes_{\mathbb{Z}}K$ over any field K is semisimple if and only if any of these numbers is not zero in K. This shows that these two numbers have the same prime factors. If moreover $\mathcal{C}$ is pivotal, it gives a numerical criterion that $\mathcal{C}$ is nondegenerate if and only if any of these numbers is not zero in $\mathbb{k}$ . For the case that $\mathcal{C}$ is a spherical fusion category over the field $\mathbb{C}$ of complex numbers, these two numbers and the Frobenius–Schur exponent of $\mathcal{C}$ share the same prime factors. This may be thought of as another version of the Cauchy theorem for spherical fusion categories.
- Published
- 2020
48. Two qualified models of learning by doing
- Author
-
Samuel Huhui
- Subjects
050208 finance ,Process (engineering) ,media_common.quotation_subject ,05 social sciences ,Regression ,Expression (mathematics) ,Learning-by-doing (economics) ,Production planning ,Learning curve ,0502 economics and business ,Prime factor ,050207 economics ,Function (engineering) ,Mathematical economics ,media_common - Abstract
Learning by doing (or learning curves) is a well-known law in economics and psychology, but no consensus has been achieved on the “qualified” models for more than a century. This article explores the expression of learning by doing in a way where the expression is not involved with changing the prime factors of a learning process. If one prime factor changes dramatically during the course of a learning process, the result of the regression is actually an approach to link two different learning curves. If the two curves are distinguishable, they each obey the law of learning by doing, which will progress rapidly at the initial phase and gradually slow down to a flat end. This article presents two functions as the law of learning by doing: The general exponential model is 0.79:0.21 better than the exponential delay model, whereas the later has the ability to investigate the change of loading factors. This ability makes the models a powerful tool for entrepreneurs and managers in investment and production planning. Key words: Learning by doing, function expression, single equation models, firm behavior, empirical analysis.
- Published
- 2020
49. Codimension 2 cycles on Severi–Brauer varieties and decomposability
- Author
-
Eoin Mackall
- Subjects
General Mathematics ,010102 general mathematics ,Algebraic geometry ,Codimension ,01 natural sciences ,Combinatorics ,Number theory ,0103 physical sciences ,Prime factor ,Exponent ,Torsion (algebra) ,010307 mathematical physics ,0101 mathematics ,Computer Science::Databases ,Mathematics - Abstract
In this text we show that one can generalize results showing that $$\mathrm {CH}^2(X)$$ , for various Severi–Brauer varieties X, is sometimes torsion free. In particular we show that for any pair of odd integers (n, m), with m dividing n and sharing the same prime factors, one can find a central simple k-algebra A of index n and exponent m that moreover has $$\mathrm {CH}^2(X)$$ torsion free for $$X=\mathrm {SB}(A)$$ . One can even take $$k={\mathbb {Q}}$$ in this construction.
- Published
- 2020
50. UNEXPECTED AVERAGE VALUES OF GENERALIZED VON MANGOLDT FUNCTIONS IN RESIDUE CLASSES
- Author
-
Arindam Roy and Nicolas Robles
- Subjects
Residue (complex analysis) ,General Mathematics ,Central object ,Liouville function ,010102 general mathematics ,Expected value ,01 natural sciences ,Riemann zeta function ,010101 applied mathematics ,Constant factor ,Combinatorics ,symbols.namesake ,Prime factor ,Elementary proof ,symbols ,0101 mathematics ,Mathematics - Abstract
In order to study integers with few prime factors, the average of $\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$ has been a central object of research. One of the more important cases, $k=2$ , was considered by Selberg [‘An elementary proof of the prime-number theorem’, Ann. of Math. (2)50 (1949), 305–313]. For $k\geq 2$ , it was studied by Bombieri [‘The asymptotic sieve’, Rend. Accad. Naz. XL (5)1(2) (1975/76), 243–269; (1977)] and later by Friedlander and Iwaniec [‘On Bombieri’s asymptotic sieve’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)5(4) (1978), 719–756], as an application of the asymptotic sieve. Let $\unicode[STIX]{x1D6EC}_{j,k}:=\unicode[STIX]{x1D707}_{j}\ast \log ^{k}$ , where $\unicode[STIX]{x1D707}_{j}$ denotes the Liouville function for $(j+1)$ -free integers, and $0$ otherwise. In this paper we evaluate the average value of $\unicode[STIX]{x1D6EC}_{j,k}$ in a residue class $n\equiv a\text{ mod }q$ , $(a,q)=1$ , uniformly on $q$ . When $j\geq 2$ , we find that the average value in a residue class differs by a constant factor from the expected value. Moreover, an explicit formula of Weil type for $\unicode[STIX]{x1D6EC}_{k}(n)$ involving the zeros of the Riemann zeta function is derived for an arbitrary compactly supported ${\mathcal{C}}^{2}$ function.
- Published
- 2020
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