• Title/Summary/Keyword: Number Theory

### ON THE MULTI-DIMENSIONAL PARTITIONS OF SMALL INTEGERS

• Kim, Jun-Kyo
• East Asian mathematical journal
• /
• v.28 no.1
• /
• pp.101-107
• /
• 2012
• For each dimension exceeds 1, determining the number of multi-dimensional partitions of a positive integer is an open question in combinatorial number theory. For n ${\leq}$ 14 and d ${\geq}$ 1 we derive a formula for the function ${\wp}_d(n)$ where ${\wp}_d(n)$ denotes the number of partitions of n arranged on a d-dimensional space. We also give an another definition of the d-dimensional partitions using the union of finite number of divisor sets of integers.

### A Proposal on Contents and Teaching-Learning Programs of Algebra Related Courses in Teachers College (교사 양성 대학에서의 대수 영역의 학습과 지도)

• 신현용
• The Mathematical Education
• /
• v.42 no.4
• /
• pp.481-501
• /
• 2003
• The main purpose of this work is to propose programs of algebra courses for the department of mathematics education of teacher training universities. Set Theory, Linear Algebra, Number Theory, Abstract Algebra I, Abstract Algebra II, and Philosophy of Mathematics for School Teachers are discussed in this article.

### An Efficient Algorithm for Performance Analysis of Multi-cell and Multi-user Wireless Communication Systems

• Wang, Aihua;Lu, Jihua
• KSII Transactions on Internet and Information Systems (TIIS)
• /
• v.5 no.11
• /
• pp.2035-2051
• /
• 2011
• Theoretical Bit Error Rate (BER) and channel capacity analysis are always of great interest to the designers of wireless communication systems. At the center of such analyses people are often encountered with a high-dimensional multiple integrals with quite complex integrands. Conventional Gaussian quadrature is inefficient in handling problems like this, as it tends to entail tremendous computational overhead, and the principal order of its error term increase rapidly with the dimension of the integral. In this paper, we propose a new approach to calculate complex multi-fold integrals based on the number theory. In contrast to Gaussian quadrature, the proposed approach requires less computational effort, and the principal order of its error term is independent of the dimension. The effectiveness of the number theory based approach is examined in BER and capacity analyses for practical systems. In particular, the results generated by numerical computation turn out in good match with that of Monte-Carlo simulations.

### On the ring of integers of cyclotomic function fields

• Bae, Sunghan;Hahn, Sang-Geun
• Bulletin of the Korean Mathematical Society
• /
• v.29 no.1
• /
• pp.153-163
• /
• 1992
• Carlitz module is used to study abelian extensions of K=$F_{q}$(T). In number theory every abelian etension of Q is contained in a cyclotomic field. Similarly every abelian extension of $F_{q}$(T) with some condition on .inf. is contained in a cyclotomic function field. Hence the study of cyclotomic function fields in analogy with cyclotomic fields is an important subject in number theory. Much are known in this direction such as ring of integers, class groups and units ([G], [G-R]). In this article we are concerned with the ring of integers in a cyclotomic function field. In [G], it is shown that the ring of integers is generated by a primitive root of the Carlitz module using the ramification theory and localization. Here we will give another proof, which is rather elementary and explicit, of this fact following the methods in [W].[W].

### A Study on the Extended RSA Public Key Cryptosystem Based on the Integral Number Theory (정수론에 근거한 확장 RSA 공개키 암호 방식에 관한 연구)

• 류재관;이지영
• Journal of the Korea Society of Computer and Information
• /
• v.3 no.2
• /
• pp.183-188
• /
• 1998
• This paper proposes an extended RSA public-key cryptosystem which extends a conventional one. The number of multiplication times has been increased by extending the modulus parameters p, q. This result shows the increase of computational complexity which required in cryptanalysis. It also improves the strength of RSA public key cryptosystem through this proof which is based on integral number theory.

### Mathematical truth and Provability (수학적 참과 증명가능성)

• Jeong, Gye-Seop
• Korean Journal of Logic
• /
• v.8 no.2
• /
• pp.3-32
• /
• 2005
• Hilbert's rational ambition to establish consistency in Number theory and mathematics in general was frustrated by the fact that the statement itself claiming consistency is undecidable within its formal system by $G\ddot{o}del's$ second theorem. Hilbert's optimism that a mathematician should not say "Ignorabimus" ("We don't know") in any mathematical problem also collapses, due to the presence of a undecidable statement that is neither provable nor refutable. The failure of his program receives more shock, because his system excludes any ambiguity and is based on only mechanical operations concerning signs and strings of signs. Above all, $G\ddot{o}del's$ theorem demonstrates the limits of formalization. Now, the notion of provability in the dimension of syntax comes to have priority over that of semantic truth in mathematics. In spite of his failure, the notion of algorithm(mechanical processe) made a direct contribution to the emergence of programming languages. Consequently, we believe that his program is failure, but a great one.

### Hong Jung Ha's Number Theory (홍정하(洪正夏)의 수론(數論))

• Hong, Sung-Sa;Hong, Young-Hee;Kim, Chang-Il
• Journal for History of Mathematics
• /
• v.24 no.4
• /
• pp.1-6
• /
• 2011
• We investigate a method to find the least common multiples of numbers in the mathematics book GuIlJib(구일집(九一集), 1724) written by the greatest mathematician Hong Jung Ha(홍정하(洪正夏), 1684~?) in Chosun dynasty and then show his achievement on Number Theory. He first noticed that for the greatest common divisor d and the least common multiple l of two natural numbers a, b, l = $a\frac{b}{d}$ = $b\frac{a}{d}$ and $\frac{a}{d}$, $\frac{b}{d}$ are relatively prime and then obtained that for natural numbers $a_1,\;a_2,{\ldots},a_n$, their greatest common divisor D and least common multiple L, $\frac{ai}{D}$($1{\leq}i{\leq}n$) are relatively prime and there are relatively prime numbers $c_i(1{\leq}i{\leq}n)$ with L = $a_ic_i(1{\leq}i{\leq}n)$. The result is one of the most prominent mathematical results Number Theory in Chosun dynasty. The purpose of this paper is to show a process for Hong Jung Ha to capture and reveal a mathematical structure in the theory.

### THE FUNDAMENTAL THEOREM OF ALGEBRA AND THE DIVERGENCE OF RECIPROCALS OF PRIMES LOOKED AT THROUGH BERGMAN SPACES

• Zeytuncu, Yunus E.
• Bulletin of the Korean Mathematical Society
• /
• v.52 no.3
• /
• pp.699-705
• /
• 2015
• Two well known facts from elementary number theory are proven by using Bergman spaces.

### REMARKS ON FINITE FIELDS

• Kang, Shin-Won
• Bulletin of the Korean Mathematical Society
• /
• v.20 no.2
• /
• pp.81-85
• /
• 1983
• It is the purpose of this paper to give some remarks on finite fields. We shall show that the little theorem of Fermat, Euler's criterion for quadratic residue mod p, and other few theorems in the number theory can be derived from the theorems in theory of finite field K=GF(p), where p is a prime. The forms of some irreducible ploynomials over K-GF(p) will be given explicitly.

### 나머지 수 체계의 부활

• 예홍진
• Journal for History of Mathematics
• /
• v.12 no.2
• /
• pp.47-54
• /
• 1999
• We introduce some historical facts on number theory, especially prime numbers and modular arithmetic. And then, with the viewpoint of computer arithmetic, residue number systems are considered as an alternate to positional number systems so that high performance and high speed computation can be achieved in a specified domain such as cryptography and digital signal processing.