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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.

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., $$\begin{aligned} {\mathfrak {B}}(x)\asymp \frac{x}{\log x}. \end{aligned}$$

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.

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.

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

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 $$\sum _{n=0}^\infty h(A,A_n)X^n/n!$$ .

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 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.

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

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.

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.