On the divisibility among power GCD and power LCM matrices on gcd-closed sets

Let [Formula: see text] and [Formula: see text] be positive integers and let [Formula: see text] be a set of [Formula: see text] distinct positive integers. For [Formula: see text], one defines [Formula: see text]. We denote by [Formula: see text] (respectively, [Formula: see text]) the [Formula: see text] matrix having the [Formula: see text]th power of the greatest common divisor (respectively, the least common multiple) of [Formula: see text] and [Formula: see text] as its [Formula: see text]-entry. In this paper, we show that for arbitrary positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], the [Formula: see text]th power matrices [Formula: see text] and [Formula: see text] are both divisible by the [Formula: see text]th power matrix [Formula: see text] if [Formula: see text] is a gcd-closed set (i.e. [Formula: see text] for all integers [Formula: see text] and [Formula: see text] with [Formula: see text]) such that [Formula: see text]. This confirms two conjectures of Shaofang Hong proposed in 2008.

Universal Evolutionary Model for Periodical Species

Real-world examples of periodical species range from cicadas, whose life cycles are large prime numbers, like 13 or 17, to bamboos, whose periods are large multiples of small primes, like 40 or even 120. The periodicity is caused by interaction of species, be it a predator-prey relationship, symbiosis, commensalism, or competition exclusion principle. We propose a simple mathematical model, which explains and models all those principles, including listed extremal cases. This rather universal, qualitative model is based on the concept of a local fitness function, where a randomly chosen new period is selected if the value of the global fitness function of the species increases. Arithmetically speaking, the different interactions are related to only four principles: given a couple of integer periods either (1) their greatest common divisor is one, (2) one of the periods is prime, (3) both periods are equal, or (4) one period is an integer multiple of the other.

GCD of Aunu Binary Polynomials of Cardinality Seven Using Extended Euclidean Algorithm

Finite fields is considered to be the most widely used algebraic structures today due to its applications in cryptography, coding theory, error correcting codes among others. This paper reports the use of extended Euclidean algorithm in computing the greatest common divisor (gcd) of Aunu binary polynomials of cardinality seven. Each class of the polynomial is permuted into pairs until all the succeeding classes are exhausted. The findings of this research reveals that the gcd of most of the pairs of the permuted classes are relatively prime. This results can be used further in constructing some cryptographic architectures that could be used in design of strong encryption schemes.

On the Degree of the GCD of Random Polynomials over a Finite Field

In this paper, we focus on the degree of the greatest common divisor ( gcd ) of random polynomials over F q . Here, F q is the finite field with q elements. Firstly, we compute the probability distribution of the degree of the gcd of random and monic polynomials with fixed degree over F q . Then, we consider the waiting time of the sequence of the degree of gcd functions. We compute its probability distribution, expectation, and variance. Finally, by considering the degree of a certain type gcd , we investigate the probability distribution of the number of rational (i.e., in F q ) roots (counted with multiplicity) of random and monic polynomials with fixed degree over F q .

Approximate greatest common divisor of several polynomials from Hankel matrices

This paper is devoted to present a new method for computing the approximate Greatest Common Divisor (GCD) of several polynomials (not pairwise) from the generalized Hankel matrix. Our approach based on the calculation of cofactors is tested for several sets of polynomials.

On the average order of the gcd-sum function over arbitrary sets of integers

Let \mathbb{N} denote the set of all positive integers and for j,n \in \mathbb{N}, let (j,n) denote their greatest common divisor. For any S\subseteq \mathbb{N}, we define P_{S}(n) to be the sum of those (j,n) \in S, where j \in \{1,2,3, \ldots, n\}. An asymptotic formula for the summatory function of P_{S}(n) is obtained in this paper which is applicable to a variety of sets S. Also the formula given by Bordellès for the summatory function of P_{\mathbb{N}}(n) can be derived from our result. Further, depending on the structure of S, the asymptotic formulae obtained from our theorem give better error terms than those deducible from a theorem of Bordellès (see Remark 4.4).

The Improvement of Elliptic Curve Factorization Method to Recover RSA’s Prime Factors

Elliptic Curve Factorization Method (ECM) is the general-purpose factoring method used in the digital computer era. It is based on the medium length of the modulus; ECM is an efficient algorithm when the length of modulus is between 40 and 50 digits. In fact, the main costs for each iteration are modular inverse, modular multiplication, modular square and greatest common divisor. However, when compared to modular multiplication and modular square, the costs of modular inverse and greatest common divisor are very high. The aim of this paper is to improve ECM in order to reduce the costs to compute both of modular inverse and greatest common divisor. The proposed method is called Fast Elliptic Curve Factorization Method (F-ECM). For every two adjacent points on the curve, only one modular inverse and one greatest common divisor will be computed. That means it implies that the costs in both of them can be split in half. Furthermore, the length of modulus in the experiment spans from 30 to 65 bits. The experimental results show that F-ECM can finish the task faster than ECM for all cases of the modulus. Furthermore, the computation time is reduced by 30 to 38 percent.

Developing a Math Textbook using realistic Mathematics Education Approach to increase elementary students’ learning motivation

This study aims to develop an appropriate and practical math textbook in the unit of the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD) using Realistic Mathematics Education (RME) in order to increase elementary students’ learning motivation. This is a Research and Development (RnD) type of study with the Plomp model. A mathematician and a teacher assessed the validity of the textbook. The practicality of the textbook was assessed by two teachers and 15 students using questionnaires. The students' motivation was assessed by the students using questionnaires as well. The results showed that the textbook was appropriate with an average of 83.32%, the respondent results from the students’ views were practical with an average of 82.33% and very practical with an average of 87.6 from the teachers’ view. This study also found that the textbook increased the students’ learning motivation by 6.45%.