• Korean Journal of Mathematics
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• v.22 no.3
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• pp.419-427
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• 2014

Let $k=\mathbb{F}_q(T)$ be the rational function field over a finite field $\mathbb{F}_q$, where $q{\equiv}1$ mod 3. In this paper, we determine asymptotic values of average of ideal class numbers of some family of cubic Kummer extensions of k.

• Journal of Educational Research in Mathematics
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• v.14 no.1
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• pp.1-17
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• 2004

According to the 7-th curriculum, irrational number should be introduced using repeating decimals in 8-th grade mathematics. To do so, the relation between rational numbers and repeating decimals such that a number is rational number if and only if it can be represented by a repeating decimal, should be examined closely Since this relation lacks clarity in some text books, irrational numbers have only slight relation with repeating decimals in those books. Furthermore, some text books introduce irrational numbers showing that $\sqrt{2}$ is not rational number, which is out of 7-th curriculum. On the other hand, if we use numeral 0 as a repetend, many results related to repeating decimals can be represented concisely. In particular, the treatments of order relation with repeating decimals in 8-th grade text books must be reconsidered.

• Bulletin of the Korean Mathematical Society
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• v.54 no.5
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• pp.1859-1871
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• 2017

We compute the Jordan constant for the group of birational automorphisms of a projective plane ${\mathbb{P}}^2_{\mathbb{k}}$, where ${\mathbb{k}}$ is either an algebraically closed field of characteristic 0, or the field of real numbers, or the field of rational numbers.

• East Asian mathematical journal
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• v.24 no.5
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• pp.477-496
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• 2008

In this paper, we consider the relations between mathematics and music, for examples rational numbers with musical scales, irrational numbers with musical scales, the golden ratio with musical compositions, the Fourier analysis with overtones. Our aim in this paper is to enhance the students' mathematical abilities by using musical materials.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1207-1220
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• 2018

The generalized hyperharmonic numbers $h^{(m)}_n(k)$ are defined by means of the multiple harmonic numbers. We show that the hyperharmonic numbers $h^{(m)}_n(k)$ satisfy certain recurrence relation which allow us to write them in terms of classical harmonic numbers. Moreover, we prove that the Euler-type sums with hyperharmonic numbers: $$S(k,m;p):=\sum\limits_{n=1}^{{\infty}}\frac{h^{(m)}_n(k)}{n^p}(p{\geq}m+1,\;k=1,2,3)$$ can be expressed as a rational linear combination of products of Riemann zeta values and harmonic numbers. This is an extension of the results of Dil [10] and $Mez{\ddot{o}}$ [19]. Some interesting new consequences and illustrative examples are considered.

• Honam Mathematical Journal
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• v.41 no.3
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• pp.569-579
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• 2019

Cusp (parabolic) points in the extended modular group ${\bar{\Gamma}}$ are basically the images of infinity under the group elements. This implies that the cusp points of ${\bar{\Gamma}}$ are just rational numbers and the set of cusp points is $Q_{\infty}=Q{\cup}\{{\infty}\}$.The Farey graph F is the graph whose set of vertices is $Q_{\infty}$ and whose edges join each pair of Farey neighbours. Each rational number x has an integer continued fraction expansion (ICF) $x=[b_1,{\cdots},b_n]$. We get a path from ${\infty}$ to x in F as $<{\infty},C_1,{\cdots},C_n>$ for each ICF. In this study, we investigate relationships between Fibonacci numbers, Farey graph, extended modular group and ICF. Also, we give a computer program that computes the geodesics, block forms and matrix represantations.

• Journal of the Korean Mathematical Society
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• v.54 no.4
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• pp.1345-1356
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• 2017

For a given number field K, we show that the ranks of elliptic curves over K are uniformly finitely bounded if and only if the weak Mordell-Weil property holds in all (some) ultrapowers $^*K$ of K. We introduce the nonstandard weak Mordell-Weil property for $^*K$ considering each Mordell-Weil group as $^*{\mathbb{Z}}$-module, where $^*{\mathbb{Z}}$ is an ultrapower of ${\mathbb{Z}}$, and we show that the nonstandard weak Mordell-Weil property is equivalent to the weak Mordell-Weil property in $^*K$. In a saturated nonstandard number field, there is a nonstandard ring of integers $^*{\mathbb{Z}}$, which is definable. We can consider definable abelian groups as $^*{\mathbb{Z}}$-modules so that the nonstandard weak Mordell-Weil property is well-defined, and we conclude that the nonstandard weak Mordell-Weil property and the weak Mordell-Weil property are equivalent. We have valuations induced from prime numbers in nonstandard rational number fields, and using these valuations, we identify two nonstandard rational numbers.

• Journal of the Korean Mathematical Society
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• v.59 no.6
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• pp.1103-1137
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• 2022

Our motivation in this note is to find equal hyperharmonic numbers of different orders. In particular, we deal with the integerness property of the difference of hyperharmonic numbers. Inspired by finiteness results from arithmetic geometry, we see that, under some extra assumption, there are only finitely many pairs of orders for two hyperharmonic numbers of fixed indices to have a certain rational difference. Moreover, using analytic techniques, we get that almost all differences are not integers. On the contrary, we also obtain that there are infinitely many order values where the corresponding differences are integers.

• Honam Mathematical Journal
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• v.45 no.3
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• pp.542-554
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• 2023

Given a real number α, the Lagrange number of α is the supremum of all real numbers L > 0 for which the inequality |α - p/q| < (Lq²)^-1 holds for infinitely many rational numbers p/q. All Lagrange numbers less than 3 can be arranged as a set {l_p/q : p/q ∈ ℚ ∩ [0, 1]} using the Farey index. The present paper considers a function C(α) devised from Sturmian words. We demonstrate that the function C(α) contains all information on Lagrange numbers less than 3. More precisely, we prove that for any real number α ∈ (0, 1], the value C(α) - C(0) is equal to the sum of all numbers 3 - l_p/q where the Farey index p/q is less than α.

• The Mathematical Education
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• v.48 no.1
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• pp.81-91
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• 2009

This paper focuses on two problems in the 10th grade mathematics, the rational zero theorem and the content(the integer divisor) of a polynomial Among 138 students participated in the problem solving, 58 of them (42 %) has used the rational zero theorem for the factorization of polynomials. However, 30 of 58 students (52 %) consider the rational zero theorem is a mathematical fake(false statement) and they only use it to get a correct answer. There are three different types in the textbooks in dealing with the content of a polynomial with integer coefficients. Computing the greatest common divisor of polynomials, some textbooks consider the content of polynomials, some do not and others suggest both methods. This also makes students confused. We suggests that a separate section of the rational zero theorem must be included in the text. As for the content of a polynomial, we consider the polynomials are contained in the polynomial ring over the rational numbers. So computing the gcd of polynomials, guide the students to give a monic(or primitive) polynomial as ail answer.