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Solution of Linear and Quadratic Equations Based on Triangular Linear Diophantine Fuzzy Numbers
Journal of Function Spaces, 2021 This paper is introducing a new concept of triangular linear Diophantine fuzzy numbers (TLDFNs) in a generic way. We first introduce the concept of TLDFNs and then study the arithmetic operations on these numbers.
Naveed Khan +4 more
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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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Solutions of the Diophantine Equations Br=Js+Jt and Cr=Js+Jt
Journal of Mathematics, 2023 Let Brr≥0, Jrr≥0, and Crr≥0 be the balancing, Jacobsthal, and Lucas balancing numbers, respectively. In this paper, the diophantine equations Br=Js+Jt and Cr=Js+Jt are completely solved.
Ahmed Gaber, Mohiedeen Ahmed
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Application of the group action approach to solving linear Diophantine equations [PDF]
Известия Саратовского университета. Новая серия: Математика. Механика. ИнформатикаThe 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 +1 more
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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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Solution to a pair of linear, two-variable, Diophantine equations with coprime coefficients from balancing and Lucas-balancing numbers [PDF]
Notes on Number Theory and Discrete Mathematics, 2023 Let Bₙ and Cₙ be the n-th balancing and Lucas-balancing numbers, respectively. We consider the Diophantine equations ax + by = (1/2)(a - 1)(b - 1) and 1 + ax + by = (1/2)(a - 1)(b - 1) for (a,b) ∈ {(Bₙ,Bₙ₊₁), (B₂ₙ₋₁,B₂ₙ₊₁), (Bₙ,Cₙ), (Cₙ,Cₙ₊₁)} and ...
R. K. Davala
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Three Diophantine equations concerning the polygonal numbers [PDF]
Notes on Number Theory and Discrete MathematicsMany 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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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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Subconvexity for additive equations: Pairs of undenary cubic forms [PDF]
, 2014 We investigate pairs of diagonal cubic equations with integral coefficients. For a class of such Diophantine systems with 11 or more variables, we are able to establish that the number of integral solutions in a large box is at least as large as the ...
Brueden, Joerg +4 more
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Geometry of the Minimal Solutions of a Linear Diophantine Equation [PDF]
SIAM Journal on Discrete Mathematics, 2021 12 pages; To appear in SIAM J.
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