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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.

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.

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 $mN^{2}$ .

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 $$n > 2$$ .

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.

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

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

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.

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).

The Piatetski-Shapiro sequences are sequences of the form $ (\left\lfloor {{n^c}} \right\rfloor)_{n = 1}^\infty $ for $ c > 1 $ and $ c \not\in \mathbb{N} $. It is conjectured that there are infinitely many primes in Piatetski-Shapiro sequences for $ c \in (1, 2) $. For every $ R \ge 1 $, we say that a natural number is an $ R $-almost prime if it has at most $ R $ prime factors, counted with multiplicity. In this paper, we prove that there are infinitely many $ R $-almost primes in Piatetski-Shapiro sequences if $ c \in (1, c_R) $ and $ c_R $ is an explicit constant depending on $ R $.

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.

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

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.

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}

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).

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

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.

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

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.

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.

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 .

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.

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

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

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

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.

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.

Modern beam shaping techniques have enabled the generation of optical fields displaying a wealth of structural features, which include three-dimensional topologies such as Möbius, ribbon strips and knots. However, unlike simpler types of structured light, the topological properties of these optical fields have hitherto remained more of a fundamental curiosity as opposed to a feature that can be applied in modern technologies. Due to their robustness against external perturbations, topological invariants in physical systems are increasingly being considered as a means to encode information. Hence, structured light with topological properties could potentially be used for such purposes. Here, we introduce the experimental realization of structures known as framed knots within optical polarization fields. We further develop a protocol in which the topological properties of framed knots are used in conjunction with prime factorization to encode information., Beam shaping methods can generate optical fields with nontrivial topologies, which are invariant against perturbations and thus interesting for information encoding. Here, the authors introduce the realization of framed optical knots to encode programs with the conjoined use of prime factorization.

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

We introduceadmissible collectionsfor a finite group 𝐺 and use them to prove that most of the finite classical groups in non-defining characteristic satisfy theQuillen dimension at 𝑝 property, a strong version of Quillen’s conjecture, at a given odd prime divisor 𝑝 of|G|\lvert G\rvert. Compared to the methods in [M. Aschbacher and S. D. Smith, On Quillen’s conjecture for the 𝑝-groups complex,Ann. of Math. (2)137(1993), 3, 473–529], our techniques are simpler.

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. If X contains a prime divisor D with rho(X)-rho(D)>2, then either X is a product of del Pezzo surfaces, or rho(X)=5 or 6. In this setting, we completely classify the case where rho(X)=5; there are 6 families, among which one is new. We also deduce the classification of Fano 4-folds with rho(X)>4 with an elementary divisorial contraction sending a divisor to a curve., 16 pages

International audience; We establish new Bombieri-Vinogradov type estimates for a wide class of multiplicative arithmetic functions and derive several applications, including: a new proof of a recent estimate by Drappeau and Topacogullari for arithmetical correlations; a theorem of Erdős-Wintner type with support equal to the level set of an additive function at shifted argument; and a law of iterated logarithm for the distribution of prime factors of integers weighted by τ (n − 1) where τ denotes the divisor function.

A famous conjecture of Hardy and Littlewood claims the subadditivity of the prime counting function, namely that ?(x+y) ? ?(x)+?(y) holds for all integers x, y ? 2, where ?(x) is the number of primes not exceeding x. It is widely believed nowadays that this conjecture is not true since Hensley and Richards stunningly discovered an incompatibility with the prime k-tuples conjecture. Despite this drawback, here we generalize the subadditivity conjecture to subsets of prime numbers possessing a rich collection of preassigned structures. We show that subadditivity holds in this extended manner over certain ranges of the parameters which are wide enough to imply that it holds in an almost all sense. Under the prime k-tuples conjecture, very large values of convex combinations of the prime counting function are obtained infinitely often, thereby indicating a strong deviation of ?(x) from being convex, even in a localized form. Finally, a Tauberian type condition is given for subsets of prime numbers which in turn implies an extension of a classical phenomenon, originally suggested by Legendre, about the asymptotically best fit functions to ?(x) of the shape x/(log x ? A)., NA

There have been several efforts to apply quantum SAT solving methods to factor large integers. While these methods may provide insight into quantum SAT solving, to date they have not led to a convincing path to integer factorization that is competitive with the best known classical method, the Number Field Sieve. Many of the techniques tried involved directly encoding multiplication to SAT or an equivalent NP-hard problem and looking for satisfying assignments of the variables representing the prime factors. The main challenge in these cases is that, to compete with the Number Field Sieve, the quantum SAT solver would need to be superpolynomially faster than classical SAT solvers. In this paper the use of SAT solvers is restricted to a smaller task related to factoring: finding smooth numbers, which is an essential step of the Number Field Sieve. We present a SAT circuit that can be given to quantum SAT solvers such as annealers in order to perform this step of factoring. If quantum SAT solvers achieve any asymptotic speedup over classical brute-force search for smooth numbers, then our factoring algorithm is faster than the classical NFS.

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

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.

Mathematics has an important role in the world of education because it is very supportive of knowledge in other fields, such as physics, chemistry, biology and science. However, most people consider mathematics to be very difficult and boring because it is abstract, so mathematics learning should be carried out in a fun and interesting manner and easily understood by students. Determining the KPK and FPB of an integer is one of the mathematical material that is often considered difficult by students in primary school, especially if the number is large enough. Usually the teacher uses a factor tree or prime factorization. It was never applied using props, so this article discusses the alternative methods of determining the KPK and FPB of an integer using the props. The visual aids used were KPK and FPB boxes. Through this teaching aid, the learning becomes more fun and students become more enthusiastic about learning and understanding the concept of material becomes more increased .

Groups of order 4 are isomorphic to either $${\mathbb {Z}}/4{\mathbb {Z}}$$ or $${\mathbb {Z}}/2{\mathbb {Z}} \times {\mathbb {Z}}/2{\mathbb {Z}}$$ . We give certain sufficient conditions permitting to specify the structure of class groups of order 4 in the family of real quadratic fields $${\mathbb {Q}}{(\sqrt{n^2+1})}$$ as n varies over positive integers. Further, we compute the values of Dedekind zeta function attached to these quadratic fields at the point $$-1$$ . As a side result, we show that the size of the class group of this family could be made as large as possible by increasing the size of the number of distinct odd prime factors of n.

We consider the isomorphism problem for groups specified by their multiplication tables. Until recently, the best published bound for the worst-case was achieved by the n^(log_p n + O(1)) generator-enumeration algorithm. In previous work with Fabian Wagner, we showed an n^((1 / 2) log_p n + O(log n / log log n)) time algorithm for testing isomorphism of p-groups by building graphs with degree bounded by p + O(1) that represent composition series for the groups and applying Luks' algorithm for testing isomorphism of bounded degree graphs. In this work, we extend this improvement to the more general class of solvable groups to obtain an n^((1 / 2) log_p n + O(log n / log log n)) time algorithm. In the case of solvable groups, the composition factors can be large which prevents previous methods from outperforming the generator-enumeration algorithm. Using Hall's theory of Sylow bases, we define a new object that generalizes the notion of a composition series with small factors but exists even when the composition factors are large. By constructing graphs that represent these objects and running Luks' algorithm, we obtain our algorithm for solvable-group isomorphism. We also extend our algorithm to compute canonical forms of solvable groups while retaining the same complexity., Comment: 22 pages. This is an updated and improved version of the results for solvable groups in arXiv:1205.0642

Let ω ( n ) denote the number of distinct prime factors of the positive integer n. In 1917, Hardy and Ramanujan showed that for all real numbers x ≥ 2 and all positive integers k, ∑ n ≤ x ω ( n ) = k 1 ≤ C x log ⁡ x ( log ⁡ log ⁡ x + D ) k − 1 ( k − 1 ) ! , where C and D are absolute constants. We derive an analogous result when the summand 1 is replaced by f ( n ) , for many nonnegative multiplicative functions f. Summing on k recovers a frequently-used mean-value theorem of Hall and Tenenbaum. We use the same idea to establish a variant of a theorem of Shirokov, concerning multiplicative functions that are o ( 1 ) on average at the primes.

For each integer m ≥ 2 , every Boolean function f can be expressed as a unique multilinear polynomial modulo m, and the degree of this multilinear polynomial is called its modulo m degree. In this paper we investigate the modulo degree complexity of total Boolean functions initiated by Parikshit Gopalan et al. [9] , in which they asked the following question: whether the degree complexity of a Boolean function is polynomially related with its modulo m degree. For m be a power of primes, it is already known that the module m degree can be arbitrarily smaller compare to the degree complexity (see Section 2 for details). When m has at least two distinct prime factors, the question remains open. Towards this question, our results include: (1) we obtain some nontrivial equivalent forms of this question; (2) we affirm this question for some special classes of functions; (3) we prove a no-go theorem, explaining why this problem is difficult to attack from the computational complexity point of view; (4) we show a super-linear separation between the degree complexity and the modulo m degree.

An arithmetical structure on the complete graph $K_n$ with $n$ vertices is given by a collection of $n$ positive integers with no common factor each of which divides their sum. We show that, for all positive integers $c$ less than a certain bound depending on $n$, there is an arithmetical structure on $K_n$ with largest value $c$. We also show that, if each prime factor of $c$ is greater than $(n+1)^2/4$, there is no arithmetical structure on $K_n$ with largest value $c$. We apply these results to study which prime numbers can occur as the largest value of an arithmetical structure on $K_n$., Comment: 10 pages; v2: minor corrections

This paper presents a new factoring technique on the modulus , where and are large prime numbers. Suppose there exists an integer satisfies the equation , for some unknown integer and is the Euler’s totient function. Our method exploits the term to be the closest integer to the unknown parameter . Hence we show that the unknown parameters and can be recovered from the list of the continued fractions expansion of Furthermore, we present an algorithm to compute the prime factors of in polynomial time after obtaining the correct tuple and.

We determine, up to multiplicative constants, the number of integers $n\leq x$ that have a divisor in $(y,2y]$ and no prime factor $\leq w$ . Our estimate is uniform in $x,y,w$ . We apply this to determine the order of the number of distinct integers in the $N\times N$ multiplication table, which are free of prime factors $\leq w$ , and the number of distinct fractions of the form $(a_{1}a_{2})/(b_{1}b_{2})$ with $1\leq a_{1}\leq b_{1}\leq N$ and $1\leq a_{2}\leq b_{2}\leq N$ .

Let $\tau(n)$ stand for the number of divisors of the positive integer $n$. We obtain upper bounds for $\tau(n)$ in terms of $\log n$ and the number of distinct prime factors of $n$.