• Honam Mathematical Journal
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• v.40 no.3
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• pp.549-559
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• 2018

In this paper, we consider the congruences involving harmonic numbers and the terms of the sequences {$U_{kn}$} and {$V_{kn}$}. For example, for an odd prime number p, $${\sum\limits_{i=1}^{p-1}}H_i{\frac{U_{k(i+m)}}{V^i_k}}{\equiv}{\frac{(-1)^kU_{k(m+1)}}{_pV^{p-1}_k}}(V^p_k-V_{kp})(mod\;p)$$, where $m{\in}{\mathbb{Z}}$ and $k{\in}{\mathbb{Z}}$ with $p{\nmid}V_k$.

• Communications of the Korean Mathematical Society
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• v.35 no.3
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• pp.745-757
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• 2020

A set {a₁, a₂, …, a_m} of positive integers is called a Diophantine m-tuple if a_ia_j + 1 is a perfect square for all 1 ≤ i < j ≤ m. In this paper, we find the structure of a torsion group of elliptic curves E_k constructed by a Diophantine triple {F_2k, F_2k+2, 4F_2k+1F_2k+2F_2k+3}, and find all integer points on the elliptic curve under assumption that rank(E_k(ℚ)) = 2.

• Journal for History of Mathematics
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• v.13 no.2
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• pp.49-64
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• 2000

Every real number can be expressed as a simple continued fraction. In particular, a number is rational if and only if its simple continued fraction has a finite number of terms. Owing to this property, continued fractions have been a powerful tool which determines a real number to be rational or not. Continued fractions provide not only a series of best estimate for a real number, but also a useful method for finding near commensurabilities between events with different periods. In this paper, we investigate the history and some properties of continued fractions, and then consider their applications in several examples. Also we explain why the Fibonacci numbers and the Golden section appear in nature in terms of continued fractions, with some examples such as the arrangements of petals round a flower, leaves round branches and seeds on seed head.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1439-1443
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• 2010

For a positive integer k $\geq$ 2, the k-bonacci sequence {$g^{(k)}_n$} is defined as: $g^{(k)}_1=\cdots=g^{(k)}_{k-2}=0$, $g^{(k)}_{k-1}=g^{(k)}_k=1$ and for n > k $\geq$ 2, $g^{(k)}_n=g^{(k)}_{n-1}+g^{(k)}_{n-2}+{\cdots}+g^{(k)}_{n-k}$. And the k-Lucas sequence {$l^{(k)}_n$} is defined as $l^{(k)}_n=g^{(k)}_{n-1}+g^{(k)}_{n+k-1}$ for $n{\geq}1$. In this paper, we give a representation of nth k-Lucas $l^{(k)}_n$ by using determinant.