Results 21 to 30 of about 4,013,096 (184)

On Some Methods for Solution of Linear Diophantine Equations

Universal Journal of Mathematics and Applications, 2020
The paper considers a linear Diophantine equation. A method (algorithm) for finding a general class of solutions of equation is proposed. The proposed algorithm is explained by examples of equations with two and three variables, trying to direct the ...
Azam Imomov, Yorqin T. Khodjaev
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Linear systems of Diophantine equations

The Electronic Journal of Linear Algebra, 2022
Given free modules $M\subseteq L$ of finite rank $f\geq 1$ over a principal ideal domain $R$, we give a procedure to construct a basis of $L$ from a basis of $M$ assuming the invariant factors or elementary divisors of $L/M$ are known. Given a matrix $A\in M_{m,n}(R)$ of rank $r$, its nullspace $L$ in $R^n$ is a free $R$-module of rank $f=n-r$.
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On the Symbolic 2-plithogenic Fermat's Non-Linear Diophantine Equation [PDF]

Neutrosophic Sets and Systems, 2023
This paper is dedicated to find all symbolic 2-plithogenic integer solutions for the symbolic 2-plithogenic Fermat's Diophantine equation.
Heba Alrawashdeh, Othman Al-Basheer, Arwa Hajjari   +2 more
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The Computing of Pythagoras Triples in Symbolic 2-Plithogenic Rings [PDF]

Neutrosophic Sets and Systems, 2023
This paper is dedicated to finding a general algorithm for generating different solutions for Pythagoras' non-linear Diophantine equation in four variables 𝑥 2 + 𝑦 2 = 𝑧 2 in symbolic 2-plithogenic rings, which are known as Pythagoras triples.
Abuobida Mohammed A. Alfahal, Yaser Ahmad Alhasan, Raja Abdullah Abdulfatah, Ahmad Abd Al-Aziz   +3 more
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On Pythagoras Triples in Symbolic 3-Plithogenic Rings [PDF]

Neutrosophic Sets and Systems, 2023
The objective of this paper is to find necessary and sufficient conditions for a symbolic 3-plithogenic triple (𝑡0 + 𝑡1𝑃1 + 𝑡2𝑃2 + 𝑡3𝑃3, 𝑠0 + 𝑠1𝑃1 + 𝑠2𝑃2 + 𝑠3𝑃3, 𝑘0 + 𝑘1𝑃1 + 𝑘2𝑃2 + 𝑘3𝑃3 ) to be a Pythagoras triple, i.e.
Abuobida Mohammed A. Alfahal, Yaser Ahmad Alhasan, Raja Abdullah Abdulfatah, Noor Edin Rabeh   +3 more
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The Frobenius Number for Jacobsthal Triples Associated with Number of Solutions

Axioms, 2023
In this paper, we find a formula for the largest integer (p-Frobenius number) such that a linear equation of non-negative integer coefficients composed of a Jacobsthal triplet has at most p representations.
Takao Komatsu, Claudio Pita-Ruiz
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Application of the group action approach to solving linear Diophantine equations [PDF]

Известия Саратовского университета. Новая серия: Математика. Механика. Информатика
The article substantiates a method for solving linear Diophantine equations using the theory of group actions. The purpose of this paper is to introduce actions of certain groups on the set of linear Diophantine equations and to study their ...
Chistov, Ivan Sergeevich, Tsybulya, Liliya Mikhailovna   +1 more
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On the Conditions for Symbolic 3-Plithogenic Pythagoras Quadruples [PDF]

Neutrosophic Sets and Systems, 2023
The objective of this paper is to find the necessary and sufficient conditions for a symbolic 3-plithogenic quadruple (𝑡0 + 𝑡1𝑃1 + 𝑡2𝑃2 + 𝑡3𝑃3, 𝑠0 + 𝑠1𝑃1 + 𝑠2𝑃2 + 𝑠3𝑃3, 𝑘0 + 𝑘1𝑃1 + 𝑘2𝑃2 + 𝑘3𝑃3, 𝑙0 + 𝑙1𝑃1 + 𝑙2𝑃2 + 𝑙3𝑃3 ) to be a Pythagoras quadruple, i.e.
Abuobida Mohammed A. Alfahal, Yaser Ahmad Alhasan, Raja Abdullah Abdulfatah, Sara Sawalmeh   +3 more
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Small solutions of linear Diophantine equations

Discrete Mathematics, 1986
Given a system of linear diophantine equations and let (*) \(Ax=B\) be its matrix form, where \(A=(a_{ij})\) is a \(m\times n\), \(x=(x_k)\) and \(B=(a_{k,n+1})\) are \(m\times 1\) matrices. Further, given integers \(1\leq j_1< \cdots < j_m\leq n+1\), let \(d_{j_1,\ldots, j_m} = \det (a_{i,j_r}),\) \(1\leq i,r\leq m\), \(X=\sup \{| d_{j_1,\ldots, j_m}|:
I. Borosh, M. Flahive, B. Treybig
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Three Diophantine equations concerning the polygonal numbers [PDF]

Notes on Number Theory and Discrete Mathematics
Many authors investigated the problem about the linear combination of two polygonal numbers being a perfect square, i.e., the Diophantine equation mPₖ(x)+nPₖ(y)=z², where Pₖ(x) denotes the x-th k-polygonal number and m, n are positive integers.
Yong Zhang, Mei Jiang, Qiongzhi Tang
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