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The function h(k) represents the smallest number m such that every sequence of m consecutive integers contains an integer coprime to the first k primes. We give a new computational method for calculating strong upper bounds on h(k). [ABSTRACT FROM AUTHOR]

We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with k prime factors for every k between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the non-uniform distribution of primes p with the property that p - 1 divides a highly composite A [ABSTRACT FROM AUTHOR]

Abstract: We show that for each non-scattered linear order the set of non-scattered subsets of ordered by the inclusion is forcing equivalent to the two-step iteration of the Sacks forcing and a -closed forcing. If the equality or PFA holds in the ground model, then the second iterand is forcing equivalent to the algebra of the Sacks extension. [Copyright &y& Elsevier]

A robust control oriented identification approach is proposed to deal with the identification of errors-in-variables models (EIVMs), which are corrupted with input and output noises. Based on normalised coprime factor model (NCFM) representations, a frequency-domain perturbed NCFM for an EIVM is derived according to a geometrical explanation for the v-gap metric. As a result, identification of the EIVM is converted into that of the NCFM. Besides an identified nominal NCFM, its worst case error has to be quantified. Unlike other traditional control-oriented identification methods, the v-gap metric is employed to measure the uncertainties including a priori information on the disturbing noises and the worst case error for the resulting nominal NCFM. Since this metric is also used as an optimisation criterion, the associate parameter estimation problem can be effectively solved by linear matrix inequalities. Finally, a numerical simulation shows the effectiveness of the proposed method. [ABSTRACT FROM PUBLISHER]

We present new computations of tight shapes obtained using the constrained gradient descent code ridgerunner for 544 composite knots with 12 and fewer crossings, expanding our dataset to 943 knots and links. We use the new data set to analyze two outstanding conjectures about tight knots, namely that the ropelengths of composite knots are at least 4π - 4 less than the sums of the prime factors and that the writhes of composite knots are the sums of the writhes of the prime factors. Our numerics support the connect sum conjecture and argue against the additivity of writhe conjecture. We also present data on the number of configurations having straight segments and highly curved kinked regions. [ABSTRACT FROM AUTHOR]

Abstract: It is proved that every sufficiently large odd integer n can be written as where , , , are primes, and x has at most two prime factors. [Copyright &y& Elsevier]

Abstract: We consider a variant of a question of N. Koblitz. For an elliptic curve which is not -isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes p such that is also a prime. We consider a variant of this question. For a newform f, without CM, of weight , on with trivial Nebentypus and with integer Fourier coefficients, let (here is the p-th-Fourier coefficient of f). We show under GRH and Artinʼs Holomorphy Conjecture that there are infinitely many p such that has at most distinct prime factors. We give examples of about hundred forms to which our theorem applies. We also show, on GRH, that the number of distinct prime factors of is of normal order and that the distribution of these values is asymptotically a Gaussian distribution (“Erdős–Kac type theorem”). [Copyright &y& Elsevier]

Let a, b, c be integers. In this paper, we prove the integer solutions of the equation axy + byz + czx = 0 satisfy max{|x|, |y|, |z|} ≤ 2 max{a, b, c} when a, b, c are odd positive integers, and when a = b = 1, c = -1, the positive integer solutions of the equation satisfy max{x, y, z} < exp(exp(exp(5))). [ABSTRACT FROM AUTHOR]

In this paper, we characterize the odd positive integers n satisfying the congruence $$\sum\nolimits_{j = 1}^{n - 1} {j^{\tfrac{{n - 1}} {2}} } \equiv 0 (mod n)$$. We show that the set of such positive integers has an asymptotic density which turns out to be slightly larger than 3/8. [ABSTRACT FROM AUTHOR]

We consider finite groups in which, for all primes p, the p-part of the length of any conjugacy class is trivial or fixed. We obtain a full description in the case in which for each prime divisor p of the order of the group there exists a noncentral conjugacy class of p-power size. [ABSTRACT FROM PUBLISHER]

In this paper the concept of soft fuzzy T´ -prefilter, soft fuzzy T´ -ultrafilter, soft fuzzy prime T´ -prelter are introduced. The concept of soft fuzzy F* space and soft fuzzy normal family are discussed. The concept of soft fuzzy C space is established. Also the process of structure compactification of soft fuzzy C space is established. [ABSTRACT FROM AUTHOR]

Homotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. For any homotopy given by a composite homotopy data triple we define a complete invariant of nanophrases. This invariant is used to show that equivalence of nanophrases under such a homotopy can be calculated just by using the homotopies given by its prime factors. [ABSTRACT FROM PUBLISHER]

Let be an elliptic curve over without complex multiplication. For each prime of good reduction, let be the order of the group of points of the reduced curve over . According to a conjecture of Koblitz, there should be infinitely many such primes such that is prime, unless there are some local obstructions predicted by the conjecture. Suppose that is a curve without local obstructions (which is the case for most elliptic curves over ). We prove in this paper that, under the GRH, there are at least primes such that has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng [20, 21] and Miri & Murty [15]. This is also the first result where the dependence on the conjectural constant appearing in Koblitz's conjecture (also called the twin prime conjecture for elliptic curves) is made explicit. This is achieved by sieving a slightly different sequence than the one of [20] and [15]. By sieving the same sequence and using Selberg's linear sieve, we can also improve the constant of Zywina [24] appearing in the upper bound for the number of primes such that is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH. [ABSTRACT FROM AUTHOR]

Let P denote an almost-prime with at most r prime factors, counted according to multiplicity, and let E( N) denote the number of natural numbers not exceeding N that are congruent to 4 modulo 24 yet cannot be represented as the sum of three squares of primes and the square of one P. Then we have E( N)≪log N. This result constitutes an improvement upon that of D. I. Tolev, who obtained the same bound, but with P in place of P. [ABSTRACT FROM AUTHOR]

We prove that there are infinitely many integers n such that the total number of prime factors of (n+h1)…(n+hκ) is exactly (1 + o(1))κ Log k. Our result even ensures us that these prime factors are fairly evenly distributed among and factor n+hi. [ABSTRACT FROM PUBLISHER]

For n > 1, let , where σσ( n) is the sum of the divisors of n. We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G( N) ≥ max( G( N/ p), G( aN)), for all prime factors p of N and each positive integer a. The proof uses Robin's and Gronwall's theorems on G( n). An alternate proof of one step depends on two properties of superabundant numbers proved using Alaoglu and Erdős's results. [ABSTRACT FROM AUTHOR]

Let k≥2 be an integer. A natural number n is called k-perfect if σ(n)=kn. For any integer r≥1, we prove that the number of odd k-perfect numbers with at most r distinct prime factors is bounded by (k−1)4r3. [ABSTRACT FROM PUBLISHER]

Let φφ( t) ∈ ℚℚ( t) have degree d ≧ 2. For a given rational number x0, define x n+1 = φφ( xn) for each n ≧ 0. If this sequence is not eventually periodic, and if φφ does not lie in one of two explicitly determined affine conjugacy classes of rational functions, then x n+1 - xn has a primitive prime factor in its numerator for all sufficiently large n. The same result holds for the exceptional maps provided that one looks for primitive prime factors in the denominator of x n+1 - xn. Hence the result for each rational function φφ of degree at least 2 implies (a new proof) that there are infinitely many primes. The question of primitive prime factors of x n+ΔΔ - xn is also discussed for ΔΔ uniformly bounded. [ABSTRACT FROM AUTHOR]

In this paper, using the theory of double coverings of cyclotomic fields, we give a formula for ${\rm dim}_{\mathbb{F}_2}{\rm H}^0(G, U_{K}/U_{K}^2)$, where K = ℚ(ζn), G = Gal(K/ℚ), 픽2 = ℤ/2ℤ and UK is the unit group of K. We explicitly determine all the cyclotomic fields K = ℚ(ζn) such that ${\rm dim}_{\mathbb{F}_2}{\rm H}^0(G,U_{K}/U_{K}^2)= 1$. Then we apply it to the unit square problem raised in [Y. Li and X. Zhang, Global unit squares and local unit squares, J. Number Theory128 (2008) 2687-2694]. In particular, we prove that the unit square problem does not hold for ℚ(ζn) if n has more than three distinct prime factors, i.e. for each odd prime p, there exists a unit, which is a square in all local fields ℚ(ζn)v with v | p but not a square in ℚ(ζn), if n has more than three distinct prime factors. [ABSTRACT FROM AUTHOR]

Abstract: The discriminants of certain polynomials related to Chebyshev polynomials factor into the product of two polynomials, one of which has coefficients that are much larger than the otherʼs. Remarkably, these polynomials of dissimilar size have “almost” the same roots, and their discriminants involve exactly the same prime factors. [Copyright &y& Elsevier]

A method is given for constructing congruent numbers with three prime factors of the form 8 k + 3. A family of such numbers is given for which the Mordell-Weil rank of their associated elliptic curves equals 2, the maximal rank and expected rank for a congruent number curve of this type. [ABSTRACT FROM AUTHOR]

Let P( n) stand for the largest prime factor of n ≥ 2 and set P(1) = 1. For each integer n ≥ 2, let δδ( n) be the distance to the nearest P( n)-smooth number, that is, to the nearest integer whose largest prime factor is no larger than that of n. We provide a heuristic argument showing that ΣΣ n≤≤ x 1/ δδ( n) = (4 log 2 - 2 + o(1)) x as x →→ ∞. Moreover, given an arbitrary real-valued arithmetic function ƒ, we study the behavior of the more general function δδƒ( n) defined by δδƒ( n) = min1≤≤ m≠≠ n, ƒ( m)≤≤ƒ( n) | n - m| for n ≥ 2, and δδƒ (1) = 1. In particular, given any positive integers a < b, we show that ΣΣ a≤≤ n< b 1/ δδƒ( n) ≥ 2( b - a)/3 and that if ƒ( n) ≥ ƒ( a) for all n ∈ [ a, b[, then one has ΣΣ a< n< b δδƒ( n) ≤≤ ( b - a) log( b - a)/(2 log 2). [ABSTRACT FROM AUTHOR]

Given an integer d≥2, a d-normal number, or simply a normal number, is an irrational number whosed-ary expansion is such that any preassigned sequence, of length k≥1, taken within this expansion occurs at the expected limiting frequency, namely 1/dk. Answering questions raised by Igor Shparlinski, we show that 0.P(2)P(3)P(4)…P(n)… and 0.P(2+1)P(3+1)P(5+1)…P(p+1)…, where P(n) stands for the largest prime factor of n, are both normal numbers. [ABSTRACT FROM AUTHOR]

The article focuses on a property of positive integers depicting the interpedendence between their construction and the form of their prime factors as established by E. A. Karatsuba and M. E. Changa. Karatsuba has discovered that the deviation of the cardinalities of sets of numbers with an odd and an even number of prime factors has the tendency to reach infinity under certain conditions on the factors. It presents several theorems which constitute the distribution of positive integers whose prime factors belong to certain subsets.

Let K be a number field of degree n over ℚ, Â be the set of integers of K that are primitive over ℚ and let I(K) be its index. The prime factors of I(K) are called common factors of indices or inessential discriminant divisors. We show that these primes divide another index i(K) previously defined by Gunji and McQuillan as i(K) = lcmθ∈Âi(θ), where i(θ) = gcdx∈ℤFθ(x) and Fθ(x) is the characteristic polynomial of θ over ℚ. It is shown that there exists θ ∈ Â such that i(K) = i(θ) and an algorithm is given for the computation of such an integer. For any prime p|i(K), an integer ρK(p) defined as the number of $\bar{\theta} \in A{/}pA$ such that p|i(θ) is investigated. It is shown that this integer determines in some cases the splitting type of p in K. Some open questions related to I(K), i(K) and ρK(p) are stated. [ABSTRACT FROM AUTHOR]

Let Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that there exist infinitely many primes of the form p = [nc] such that p + 2 = Pr, where r is the least positive integer satisfying certain inequalities. In particular for $1 < c \leq \frac{30}{29} = 1.0344 \ldots$ we have r = 5. This result constitutes an improvement upon that of T. P. Peneva. [ABSTRACT FROM AUTHOR]

A frequency-domain method is proposed to cope with errors-in-variables model (EIVM) identification when the input and output noises are bounded by a certain upper bound. Based on normalised coprime factor model (NCFM) description, L2-optimal approximate models for an EIVM are first established, which consist of a system NCFM and its complementary inner factor model (CIFM) characterising the noises. Then the v-gap metric criterion is minimised to optimise a system coprime factor model, from which the system NCFM can be obtained by normalisation. During the optimisation, a priori information on the system poles can be fully used to reduce the overfitting effect caused by the noises. The associated noise CIFM can be readily constructed from the resulting estimated system NCFM by a model transformation. Compared with related identification methods, the system model can be effectively solved by linear matrix inequalities and the associated noise model can then be directly built. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: We give a direct construction for 20-regular cyclic difference packings ʼs when is a prime. Then, recursively, we prove the existence of an optimal optical orthogonal code for every nonnegative integer α and any positive integer u whose prime factors are all congruent to 1 (mod 6). [Copyright &y& Elsevier]

Abstract: This work is concerned with the prime factor decomposition (PFD) of strong product graphs. A new quasi-linear time algorithm for the PFD with respect to the strong product for arbitrary, finite, connected, undirected graphs is derived. Moreover, since most graphs are prime although they can have a product-like structure, also known as approximate graph products, the practical application of the well-known “classical” prime factorization algorithm is strictly limited. This new PFD algorithm is based on a local approach that covers a graph by small factorizable subgraphs and then utilizes this information to derive the global factors. Therefore, we can take advantage of this approach and derive in addition a method for the recognition of approximate graph products. [Copyright &y& Elsevier]

Let t(e)(n) denote the number of e-squarefree e-divisor of n. The aim of this paper is to establish a short interval result for the function t(e)(n). This enriches the properties of the e-squarefree e-divisor function. [ABSTRACT FROM AUTHOR]

Two numbers m and n are considered amicable if the sum of their proper divisors, s( n) and s( m), satisfy s( n) == m and s( m) == n. In 1981, Pomerance showed that the sum of the reciprocals of all such numbers, P, is a constant. We obtain both a lower and an upper bound on the value of P. [ABSTRACT FROM AUTHOR]

Abstract: We estimate the sizes of the sumset and the productset in the special case that , the set of positive integers free of prime factors exceeding y. [Copyright &y& Elsevier]

Let N be a sufficiently large even integer. Let p denote a prime and P denote an almost prime with at most two prime factors. In this paper, it is proved that the equation N = p + P ( p ≤ N) is solvable. [ABSTRACT FROM AUTHOR]

Abstract: Many cryptosystems are based on the factorization of large integers. The complexity of this type of factorization is still an advantage for data security developers and a challenge for both mathematicians and cryptanalysts. The security of RSA relies on the difficulty of factoring large integers. The factorization was studied earlier by old civilizations like the Greek, but their methods were extended after the emergence of computers. The paradox of RSA is that, in order to make RSA more efficient, we use a modulus , which is as small as possible. On the other hand, it is sufficient to factor in order to decrypt the encrypted messages. In this paper, we propose a new factorization method based on the square root approximation. This method allows in reducing the search for candidate prime factors of a given integer by approximating each prime factor before considering it as a candidate. [Copyright &y& Elsevier]

Abstract: The cross-product technique, introduced by Even and Litman (1992) , is extended into a full decomposition theory enabling a unique (up to isomorphism) and polynomial factorization of layered interconnection networks (including many well-known networks) into a product of prime factors. A polynomial algorithm is provided for checking whether a given layered interconnection network is isomorphic to a network that is uniquely decomposable into prime factors. [Copyright &y& Elsevier]

Fix αα > 0, and sample N integers uniformly at random from {1, 2, . . . , ⌊⌊ eααN⌋⌋}. Given ηη > 0, the probability that the maximum of the pairwise GCDs lies between N2-- ηη and N2++ ηη converges to 1 as N →→ ∞∞. More precise estimates are obtained. This is a Birthday Problem: two of the random integers are likely to share some prime factor of order N2/log( N). The proof generalizes to any arithmetical semigroup where a suitable form of the prime number theorem is valid. [ABSTRACT FROM AUTHOR]

The aim of this article is to establish some interesting inequalities involving arithmetic functions. [ABSTRACT FROM AUTHOR]

Given a ±1 random completely multiplicative function f, we prove by estimating moments that the limiting distribution of the normalized sum \begin{linenomath}\[\frac{1}{\sqrt{\Var}}{\sum_{n < x}}^r f(n)\]\end{linenomath} converges to the standard Gaussian distribution as x → ∞ when r restricts summation to n having o(log log log x) prime factors. We also give an upper bound for the large deviations of \begin{linenomath}\[{\sum_{n < x}}^k f(n),\]\end{linenomath} with the sum restricted to numbers having a fixed number k of prime factors. [ABSTRACT FROM AUTHOR]

recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has $${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})$$. We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices A, B are congruent modulo p then the characteristic polynomials of A and B are congruent modulo p, and then we show that Arnold's conjecture follows from it easily. Using this result, we prove the following generalization of Euler's theorem for any 2 × 2 integral matrix A: the characteristic polynomials of A and A are congruent modulo n. Here ϕ is the Euler function, $$\prod_{i=1}^{l} p_i^{\alpha_i}$$ is a prime factorization of n and $$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$$. [ABSTRACT FROM AUTHOR]

By a result of McKenzie [7] all finite directed graphs that satisfy certain connectivity conditions have unique prime factorizations with respect to the cardinal product. McKenzie does not provide an algorithm, and even up to now no polynomial algorithm that factors all graphs satisfying McKenzie's conditions is known. Only partial results [1, 3, 5] have been published, all of which depend on certain thinness conditions of the graphs to be factored. In this paper we weaken the thinness conditions and thus significantly extend the class of graphs for which the prime factorization can be found in polynomial time. [ABSTRACT FROM AUTHOR]

Taking a combinatorial point of view on cyclotomic polynomials leads to a larger class of polynomials we shall call the inclusion-exclusion polynomials. This gives a more appropriate setting for certain types of questions about the coefficients of these polynomials. After establishing some basic properties of inclusion-exclusion polynomials we turn to a detailed study of the structure of ternary inclusion-exclusion polynomials. The latter subclass is exemplified by cyclotomic polynomials ΦΦ pqr, where p < q < r are odd primes. Our main result is that the set of coefficients of ΦΦ pqr is simply a string of consecutive integers which depends only on the residue class of r modulo pq. [ABSTRACT FROM AUTHOR]

An almost square of type 2 is an integer n that can be factored in two different ways as n == a1 b1 == a2 b2 with a1, a2, b1, . In this paper, we continue the study of almost squares of type 2 in short intervals and improve the 1/2 upper bound. We also draw connections with almost squares of type 1. [ABSTRACT FROM AUTHOR]

Under the Riemann hypothesis and the conjecture that the order of growth of the argument of ζ(1/2 + it) is bounded by $\left( {\log t} \right)^{\frac{1} {2} + o\left( 1 \right)}$ , we show that for any given α > 0 the interval $(X,X + \sqrt X (\log X)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + o\left( 1 \right)} ]$ contains an integer having no prime factor exceeding X for all X sufficiently large. [ABSTRACT FROM AUTHOR]

The article discusses the equation for the number of positive divisors of the natural number. Noted that a natural number that is greater than 1 is harmonic provided that the number of positive divisors of the natural number and its sums, has prime factor decomposition. Illustrated is the formula of a natural number with multiplicative functions. In addition, two theorems are detailed concerning conditions wherein a natural number is lesser and/or equal to the cube root of a natural number.

In a paper published in 1993, Erdös proved that if n! = a! b!, where 1 < a ≤ b, then the difference between n and b does not exceed 5 log log n for large enough n. In the present paper, we improve this upper bound to ((1 + ε)/ log 2) log log n and generalize it to the equation a! a! ... a! = n!. In a recent paper, F. Luca proved that n − b = 1 for large enough n provided that the ABC-hypothesis holds. [ABSTRACT FROM AUTHOR]

We demonstrate that, under suitable local conditions on a finite collection F1,…, Fg of binary irreducible forms with integer coefficients, the product F1(x)·…·Fg(x) will have at most r prime factors for infinitely many x. We give explicit upper bounds for r that depend only on g and on the total degree of the product polynomial. [ABSTRACT FROM PUBLISHER]

An integer n is said to be y-friable if its largest prime factor P( n) does not exceed y. By convention, P(1)≔ 1. Classical notations are S( x, y) ≔ { n ≦ x : P( n) ≦ y} for the set of y-friable integers not exceeding x and Ψ( x, y) for its cardinality. The study of friable restrictions of arithmetic functions is closely connected to the Kubilius model of probabilistic number theory. In this framework, a variance analysis constitutes an essential feature of the probabilistic description of an arithmetical function ƒ as a random variable over S( x, y). The case of additive functions is particularly interesting: by comparing, uniformly in ƒ, the semi-empirical variance to the actual variance ( Zƒ, x, y) of a probabilistic model Zƒ, x, y, we get a quantitative measure of the discrepancy between probabilistic number theory and probability theory. In this direction, La Bretèche and Tenenbaum recently showed that, for any given c > 0, is finite and uniformly bounded in the domain c log x ≦ y ≦ x, thus extending the classical Turán-Kubilius inequality, which corresponds to the case x = y. Moreover, they also prove, in accord with Kubilius' model, that We determine the exact value of for all u ≧ 1 and provide an asymptotic formula for this quantity as u → ∞. Refining a method due to Hildebrand, we develop a new approach, resting upon the theory of self-adjoint operators in Hilbert spaces. [ABSTRACT FROM AUTHOR]

An interesting 2-adic property of the Stirling numbers of the second kind S( n, k) was conjectured by the author in 1994 and proved by De Wannemacker in 2005: νν2( S(2 n, k)) == d2( k) -- 1, 1 ≤≤ k ≤≤ 2 n. It was later generalized to νν2( S( c2 n, k)) == d2( k) -- 1, 1 ≤≤ k ≤≤ 2 n, c ≥≥ 1 by the author in 2009. Here we provide full and two partial alternative proofs of the generalized version. The proofs are based on non-standard recurrence relations for S( n, k) in the second parameter and congruential identities. [ABSTRACT FROM AUTHOR]

An overpartition of the nonnegative integer n is a non-increasing sequence of natural numbers whose sum is n in which the first occurrence of a number may be overlined. Let k ≥≥ 1 be an integer. An overpartition k-tuple of a positive integer n is a k-tuple of overpartitions wherein all listed parts sum to n. Let be the number of overpartition k-tuples of n. In this paper, we will give a short proof of Keister, Sellers and Vary's theorem on congruences for modulo powers of 2. We also obtain some congruences for modulo prime and integer 2 k. [ABSTRACT FROM AUTHOR]