• Bulletin of the Korean Mathematical Society
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• v.60 no.2
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• pp.315-323
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• 2023

First, we recall the results for prime-producing polynomials related to class number one problem of quadratic fields. Next, we give the relation between prime-producing cubic polynomials and class number one problem of the simplest cubic fields and then present the conjecture for the relations. Finally, we numerically compare the ratios producing prime values for several polynomials in some interval.

• School Mathematics
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• v.12 no.3
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• pp.371-388
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• 2010

The goals of this study are to inquire middle school students' understanding about prime number and to propose pedagogical implications for school mathematics. Written questionnaire were given to 198 Korean seventh graders who had just finished learning about prime number and prime factorization and then 20 students participated in individual interviews for member checks. In defining prime and composite numbers, the students focused on distinguishing one from another by numbering of factors of agiven natural number. However, they hardly recognize the mathematical connection between prime and composite numbers related on the multiplicative structure of natural number. This study suggests that it is needed to emphasize the conceptual relationship between divisibility and prime decomposition and the prime numbers as the multiplicative building blocks of natural numbers based on the Fundamental Theorem of Arithmetic.

• Communications of Mathematical Education
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• v.33 no.3
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• pp.255-273
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• 2019

In order to help students to understand the essence of prime concepts, this study looked at the history of prime concept development and analyzed how to introduce the concept of textbooks. In ancient Greece, primes were multiplicative atoms. At that time, the unit was not a number, but the development of decimal representations led to the integration of the unit into the number, which raised the issue of primality of 1. Based on the uniqueness of factorization into prime factor, 1 was excluded from the prime, and after that, the concept of prime of the atomic context and the irreducible concept of the divisor context are established. The history of the development of prime concepts clearly reveals that the fact that prime is the multiplicative atom is the essence of the concept. As a result of analyzing the textbooks, the textbook has problems of not introducing the concept essence by introducing the concept of prime into a shaped perspectives or using game, and the problem that the transition to analytic concept definition is radical after the introduction of the concept. Based on the results of the analysis, we have provided several pedagogical implications for helping to focus on a conceptual aspect of prime number.

• Kyungpook Mathematical Journal
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• v.45 no.3
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• pp.395-404
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• 2005

Let α be a complex number with 𝕽α > 0. Let the functions f and g be analytic in the unit disc E = {z : |z| < 1} and normalized by the conditions f(0) = g(0) = 0, f'(0) = g'(0) = 1. In the present article, we study the differential subordinations of the forms $${\alpha}{\frac{z^2f^{{\prime}{\prime}}(z)}{f(z)}}+{\frac{zf^{\prime}(z)}{f(z)}}{\prec}{\alpha}{\frac{z^2g^{{\prime}{\prime}}(z)}{g(z)}}+{\frac{zg^{\prime}(z)}{g(z)}},\;z{\in}E,$$ and $${\frac{z^2f^{{\prime}{\prime}}(z)}{f(z)}}{\prec}{\frac{z^2g^{{\prime}{\prime}}(z)}{g(z)}},\;z{\in}E.$$ As consequences, we obtain a number of sufficient conditions for star likeness of analytic maps in the unit disc. Here, the symbol ' ${\prec}$ ' stands for subordination

• Journal of the Korean Mathematical Society
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• v.52 no.4
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• pp.751-763
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• 2015

Let ${\alpha}$ be a positive integer, and let $p_1$, $p_2$ be two distinct prime numbers with $p_1$ < $p_2$. By using elementary methods, we give two equivalent conditions of all even near-perfect numbers in the form $2^{\alpha}p_1p_2$ and $2^{\alpha}p_1^2p_2$, and obtain a lot of new near-perfect numbers which involve some special kinds of prime number pairs. One kind is exactly the new Mersenne conjecture's prime number pair. Another kind has the form $p_1=2^{{\alpha}+1}-1$ and $p_2={\frac{p^2_1+p_1+1}{3}}$, where the former is a Mersenne prime and the latter's behavior is very much like a Fermat number.

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.41-46
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• 2016

A graph G of order n has prime cordial labeling if its vertices can be assigned the distinct labels 1, $2{\cdots}$, n such that if each edge xy in G is assigned the label 1 in case the labels of x and y are relatively prime and 0 otherwise, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. In this paper, we give a complete characterization of complete graphs which are prime cordial and we give a prime cordial labeling of the closed helm ${\bar{H}}_n$, and present a new way of prime cordial labeling of $P^2_n$. Finally we make a correction of the proof of Theorem 2.5 in [12].

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1377-1385
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• 2021

In this paper, we consider a general number system with a base m in order to determine if a positive integer x is prime. We show that the base m providing the most efficient test is the primorial p_n# when p_n# < x < p_n+1# and establish a necessary and sufficient condition for x in between consecutive primorials to be determined as a prime number.

• Korean Journal of Mathematics
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• v.29 no.4
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• pp.733-740
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• 2021

Let (F_n)_n≥0 be the Fibonacci sequence and p be a prime number. For 1≤k≤m, the Fibonomial coefficient is defined as $$\[\array{m\\k}\]_F=\frac{F_{m-k+1}{\ldots}{F_{m-1}F_m}}{{F_1}{\ldots}{F_k}}$$ and $\[\array{m\\k}\]_F=0$ whan k>m. Let a and n be positive integers. In this paper, we find the conditions of prime number p which divides Fibonomial coefficient $\[\array{P^{a+n}\\{p^a}}\]_F$. Furthermore, we also find the conditions of p when $\[\array{P^{a+n}\\{p^a}}\]_F$ is not divisible by p.

• The Transactions of the Korea Information Processing Society
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• v.7 no.9
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• pp.2913-2919
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• 2000

It is essential to get large prime numbers in the design of asymmetric encryption algorithm. However, the pseudoprime numbers with high possibility to be primes have been generally used in the asymmetric encryption algorithms, because it is very difficult to find large deterministic prime numbers. In this paper, we propose a new method of deterministic prime number generation. The prime numbers generated by the proposed method have a 100% precise prime characteristic. They are also guaranteed reliability, security strength, and an ability of primitive element generation.

• Journal of applied mathematics & informatics
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• v.34 no.3_4
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• pp.227-237
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• 2016

In this paper we introduce a new graph labeling called k-prime cordial labeling. Let G be a (p, q) graph and 2 ≤ p ≤ k. Let f : V (G) → {1, 2, . . . , k} be a map. For each edge uv, assign the label gcd (f(u), f(v)). f is called a k-prime cordial labeling of G if |v_f (i) − v_f (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |e_f (0) − e_f (1)| ≤ 1 where v_f (x) denotes the number of vertices labeled with x, e_f (1) and e_f (0) respectively denote the number of edges labeled with 1 and not labeled with 1. A graph with a k-prime cordial labeling is called a k-prime cordial graph. In this paper we investigate the k-prime cordial labeling behavior of a star and we have proved that every graph is a subgraph of a k-prime cordial graph. Also we investigate the 3-prime cordial labeling behavior of path, cycle, complete graph, wheel, comb and some more standard graphs.