• Title, Summary, Keyword: Diophantine equations

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IRREDUCIBILITY OF POLYNOMIALS AND DIOPHANTINE EQUATIONS

  • Woo, Sung-Sik
    • 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.

ABS ALGORITHM FOR SOLVING A CLASS OF LINEAR DIOPHANTINE INEQUALITIES AND INTEGER LP PROBLEMS

  • Gao, Cheng-Zhi;Dong, Yu-Lin
    • 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.

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The Diophantine Equation ax6 + by3 + cz2 = 0 in Gaussian Integers

  • IZADI, FARZALI;KHOSHNAM, FOAD
    • 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.

FIBONACCI AND LUCAS NUMBERS ASSOCIATED WITH BROCARD-RAMANUJAN EQUATION

  • Pongsriiam, Prapanpong
    • 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.

디오판틴 방정식의 해들에 대한 연산 및 성질 연구

  • Lyou, Ik-Seung;Kim, Jeong-Soo;Kim, Yeoun-Ho;Kim, Hyeung-Kyun
    • 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.

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CONTINUED FRACTION AND DIOPHANTINE EQUATION

  • Gadri, Wiem;Mkaouar, Mohamed
    • 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.

GENERALIZED FIBONACCI AND LUCAS NUMBERS OF THE FORM wx2 AND wx2 ∓ 1

  • Keskin, Refik
    • 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.