• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.93-107
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• 2005

Based on the recently developed ABS algorithm for solving linear Diophantine equations, we present a special ABS algorithm for solving such equations which is effective in computation and storage, not requiring the computation of the greatest common divisor. A class of equations always solvable in integers is identified. Using this result, we discuss the ILP problem with upper and lower bounds on the variables.

• Bulletin of the Korean Mathematical Society
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• v.51 no.5
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• pp.1369-1374
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• 2014

We show that the Diophantine equation $$aQ(x_1,x_2)+bQ(x_3,x_4)+cQ(x_5,x_6)=abc$$ has integral solutions for arbitrary positive integers a, b, c when Q(x, y) is a norm form for some imaginary quadratic fields.

• Journal of the Korean Mathematical Society
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• v.47 no.1
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• pp.101-112
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• 2010

In [3] we showed that a polynomial over a Noetherian ring is divisible by some other polynomial by looking at the matrix formed by the coefficients of the polynomials which we called the resultant matrix. In this paper, we consider the polynomials with coefficients in a field and divisibility of a polynomial by a polynomial with a certain degree is equivalent to the existence of common solution to a system of Diophantine equations. As an application we construct a family of irreducible quartics over $\mathbb{Q}$ which are not of Eisenstein type.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.349-353
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• 2008

Using the recently developed ABS algorithm for solving linear Diophantine equations we introduce an algorithm for solving a system of m linear integer inequalities in n variables, m $\leq$ n, with full rank coefficient matrix. We apply this result to solve linear integer programming problems with m $\leq$ n inequalities.

• Kyungpook Mathematical Journal
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• v.55 no.3
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• pp.587-595
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• 2015

In this article, we will examine the Diophantine equation $ax^6+by^3+cz^2=0$, for arbitrary rational integers a, b, and c in Gaussian integers and find all the solutions of this equation for many different values of a, b, and c. Moreover, two equations of the type $x^6{\pm}iy^3+z^2=0$, and $x^6+y^3{\pm}wz^2=0$ are also discussed, where i is the imaginary unit and w is a third root of unity.

• Korean Journal of Mathematics
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• v.28 no.3
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• pp.539-554
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• 2020

A set {a₁, a₂, …, a_m} of m distinct positive integers is called a D(-1)-m-tuple if the product of any distinct two elements in the set decreased by one is a perfect square. In this paper, we find a solution of Pellian equations which is constructed by D(-1)-triples and using this result, we prove the extendibility of D(-1)-pair with some conditions.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.699-709
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• 2016

Our paper is devoted to the study of certain diophantine equations on the ring of polynomials over a finite field, which are intimately related to algebraic formal power series which have partial quotients of unbounded degree in their continued fraction expansion. In particular it is shown that there are Pisot formal power series with degree greater than 2, having infinitely many large partial quotients in their simple continued fraction expansions. This generalizes an earlier result of Baum and Sweet for algebraic formal power series.

• Communications of the Korean Mathematical Society
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• v.32 no.3
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• pp.511-522
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• 2017

We explicitly solve the diophantine equations of the form $$A_{n_1}A_{n_2}{\cdots}A_{n_k}{\pm}1=B^2_m$$, where $(A_n)_{n{\geq}0}$ and $(B_m)_{m{\geq}0}$ are either the Fibonacci sequence or Lucas sequence. This extends the result of D. Marques [9] and L. Szalay [13] concerning a variant of Brocard-Ramanujan equation.

• East Asian mathematical journal
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• v.23 no.3
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• pp.371-380
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• 2007

It is well known that the solutions of the Diophantine equation $x^2+xy-y^2=1$ is related to the Fibonacci sequence. In this study, we generalize the above fact to the tribonacci sequence and its generalized from using the group structure of solutions of some Diophantine equations.

• Bulletin of the Korean Mathematical Society
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• v.51 no.4
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• pp.1041-1054
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• 2014

Let $P{\geq}3$ be an integer and let ($U_n$) and ($V_n$) denote generalized Fibonacci and Lucas sequences defined by $U_0=0$, $U_1=1$; $V_0= 2$, $V_1=P$, and $U_{n+1}=PU_n-U_{n-1}$, $V_{n+1}=PV_n-V_{n-1}$ for $n{\geq}1$. In this study, when P is odd, we solve the equations $V_n=kx^2$ and $V_n=2kx^2$ with k | P and k > 1. Then, when k | P and k > 1, we solve some other equations such as $U_n=kx^2$, $U_n=2kx^2$, $U_n=3kx^2$, $V_n=kx^2{\mp}1$, $V_n=2kx^2{\mp}1$, and $U_n=kx^2{\mp}1$. Moreover, when P is odd, we solve the equations $V_n=wx^2+1$ and $V_n=wx^2-1$ for w = 2, 3, 6. After that, we solve some Diophantine equations.