• Honam Mathematical Journal
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• v.42 no.1
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• pp.139-150
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• 2020

Let p be a prime number with p > 3, p ≡ 3 (mod 4) and let n be a positive integer. In this paper, we prove that the Diophantine equation (5pn² - 1)^x + (p(p - 5)n² + 1)^y = (pn)^z has only the positive integer solution (x, y, z) = (1, 1, 2) where pn ≡ ±1 (mod 5). As an another result, we show that the Diophantine equation (35n² - 1)^x + (14n² + 1)^y = (7n)^z has only the positive integer solution (x, y, z) = (1, 1, 2) where n ≡ ±3 (mod 5) or 5 | n. On the proofs, we use the properties of Jacobi symbol and Baker's method.

• 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 Chungcheong Mathematical Society
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• v.34 no.3
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• pp.209-219
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• 2021

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. Let F_n be the nth Fibonacci number which is defined by F₀ = 0, F₁ = 1 and F_n+2 = F_n+1 + F_n. In this paper, we find the extendibility of Diophantine pairs {F_2k, b} with some conditions.

• 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.

• Journal of the Korean Mathematical Society
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• v.55 no.2
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• pp.425-445
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• 2018

A set {$a_1,\;a_2,{\ldots},\;a_m$} of positive integers is called a Diophantine m-tuple if $a_ia_j+1$ is a perfect square for all $1{\leq}i$ < $j{\leq}m$. In this paper, we show that the form of third element in Diophantine pairs and develop some results which are needed to prove the extendibility of the Diophantine pair {a, b} with some conditions. By using this result, we prove the extendibility of Diophantine pairs {$F_{k-2}F_{k+1},\;F_{k-1}F_{k+2}$} and {$F_{k-2}F_{k-1},\;F_{k+1}F_{k+2}$}, where $F_n$ is the n-th Fibonacci number.

• 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.

• Journal for History of Mathematics
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• v.21 no.3
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• pp.113-126
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• 2008

This work is devoted to study about the abc conjecture: how it works in the development of the proof of Fermat's last theorem and more generally in the diophantine equation theory. And it is also studied the application of the abc conjecture to the iteration of polynomials.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1133-1138
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• 2015

In 1956, $Je{\acute{s}}manowicz$ conjectured that, for any positive integer n and any primitive Pythagorean triple (a, b, c) with $a^2+b^2=c^2$, the equation $(an)^x+(bn)^y=(cn)^z$ has the unique solution (x, y, z) = (2, 2, 2). In this paper, under some conditions, we prove the conjecture for the primitive Pythagorean triples $(a,\;b,\;c)=(4k^2-1,\;4k,\;4k^2+1)$.

• Kyungpook Mathematical Journal
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• v.63 no.3
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• pp.345-353
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• 2023

Reduced solutions of size 2 and degree n of the Tarry-Escott problem over M₂(ℚ) are determined. As an application, the diophantine equation αAⁿ + βBⁿ = αCⁿ + βDⁿ, where α, β are rational numbers satisfying α + β ≠ 0 and n ∈ {1, 2}, is completely solved for A, B, C, D ∈ M₂(ℚ).

• Journal of the Chungcheong Mathematical Society
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• v.5 no.1
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• pp.57-63
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• 1992