• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.37-51
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• 2003

Enumeration of the primes with difference 4 between consecutive primes, is counted up to 5${\times}$10$\^$10/, yielding the counting function ,r2,4(5${\times}$10$\^$10/) = l18905303. The sum of reciprocals of primes with gap 4 between consecutive primes is computed B$_4$(5 ${\times}$ 10$\^$10/) = 1.1970s4473029 and B$_4$ = 1.197054 ${\pm}$ 7 ${\times}$ 10$\^$-6/. And Enumeration of the primes with difference 6 between consecutive primes, is counted up to 5${\times}$10$\^$10/, yielding the counting function $\pi$$\_$2.6/(5${\times}$10$\^$10/) = 215868063. The sum of reciprocals of primes with gap 6 between consecutive primes is computed B$\_$6/(5${\times}$10$\^$10/) = 0.93087506039231 and B$\_$6/ = 1.135835 ${\pm}$ 1.2${\times}$10$\^$-6/.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.441-452
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• 2006

We introduce the counting function ${\pi}^*_{2.8}(x)$ of the primes with difference 8 between consecutive primes ($p_n,\;p_{n+l}=p_n+8$) can be approximated by logarithm integral $Li^*_{2.8}$. We calculate the values of ${\pi}^*_{2.8}(x)$ and the sum $C_{2,8}(x)$ of reciprocals of primes with difference 8 between consecutive primes $p_n,\;p_{n+l}=p_n+8$ where x is counted up to $7{\times}10^{10}$. From the results of these calculations. we obtain ${\pi}^*_{2.8}(7{\times}10^{10}$)= 133295081 and $C_{2.8}(7{\times}10^{10}) = 0.3374{\pm}2.6{\times}10^{-4}$.

• Journal of the Korean Mathematical Society
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• v.59 no.2
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• pp.407-420
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• 2022

It is proved that every sufficiently large even integer can be represented as a sum of two squares of primes, two cubes of primes, two fourth powers of primes and 17 powers of 2.

• Journal of Internet of Things and Convergence
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• v.6 no.4
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• pp.49-57
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• 2020

In this paper, I provided an approximation function Li*2,10(x) using logarithm integral for the counting function π*2,10(x) of consecutive deca primes. Several personal computers and Mathematica were used to validate the approximation function Li*2,10(x). I found the real value of π*2,10(x) and approximate value of Li*2,10(x) for various x ≤ 1011. By the result of theses calculations, most of the error rates are margins of error of 0.005%. Also, I proved that the sum C2,10(∞) of reciprocals of all primes with difference 10 between primes is finite. To find C2,10(∞), I computed the sum C2,10(x) of reciprocals of all consecutive deca primes for various x ≤ 1011 and I estimate that C2,10(∞) probably lies in the range C2,10(∞)=0.4176±2.1×10-3.

• Korean Journal of Mathematics
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• v.24 no.1
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• pp.107-138
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• 2016

Let $a{\geqslant}2$ be a xed integer in this paper. By using the method of Goldston, Pintz and Yildirm, we will prove that there are innitely many almost-primes which can be represented as $p+a^m$ in at least two dierent ways.

• Journal of Integrative Natural Science
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• v.14 no.4
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• pp.159-161
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• 2021

In this article, we will study which prime numbers are represented by the principal form. p = x² + ny². We will also give some results related to the form of primes of the form x² + axy + y².

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.83-96
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• 2013

Let K be a number field and fix a prime number $p$. For any set S of primes of K, we here say that an elliptic curve E over K has S-reduction if E has bad reduction only at the primes of S. There exists the set $B_{K,p}$ of primes of K satisfying that any elliptic curve over K with $B_{K,p}$-reduction has no $p$-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over K with $B_{K,p}$-reduction and a $p$-torsion point. The action of the absolute Galois group on the $p$-torsion subgroup of E gives its associated Galois representation $\bar{\rho}_{E,p}$ modulo $p$. We also study the irreducibility and surjectivity of $\bar{\rho}_{E,p}$ for semistable elliptic curves with $B_{K,p}$-reduction.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.89-96
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• 2009

In this article, we show a generalization of Clement's theorem on the pair of primes. For any integers n and k, integers n and n + 2k are a pair of primes if and only if 2k(2k)![(n - 1)! + 1] + ((2k)! - 1)n ${\equiv}$ 0 (mod n(n + 2k)) whenever (n, (2k)!) = (n + 2k, (2k)!) = 1. Especially, n or n + 2k is a composite number, a pair (n, n + 2k), for which 2k(2k)![(n - 1)! + 1] + ((2k)! - 1)n ${\equiv}$ 0 (mod n(n + 2k)) is called a pair of pseudoprimes for any positive integer k. We have pairs of pseudorimes (n, n + 2k) with $n{\leq}5{\times}10^4$ for each positive integer $k(4{\leq}k{\leq}10)$.

• Journal of the Korean Mathematical Society
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• v.36 no.5
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• pp.847-853
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• 1999

The purpose of this paper is to develop the theory of coassociated primes and to investigate Melkersson's question [8].

• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.251-258
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• 2002

Let R be a commutative Noetherian ring and M an R-module. In this paper we will consider the asymptotic behaviour of ideals relative to an R-module E which is M-injective.