On the Exponential Diophantine Equation (13&lt;sup&gt;2m&lt;/sup&gt;) + (6r + 1)&lt;sup&gt;n&lt;/sup&gt; = z&lt;sup&gt;2&lt;/sup&gt;

Aggarwal, Sudhanshu; Kumar, S. Dinesh

doi:10.3329/jsr.v13i3.52611

Cited by 5 publications

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References 5 publications

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“…The Diophantine equation 193 x + 211 y = z 2 was studied by Aggarwal [5]. Aggarwal and Kumar [6] examined the exponential Diophantine equation (13 2m ) + (6r + 1) n = z 2 . Aggarwal and Upadhyaya [7] studied the Diophantine equation 8 α + 67 β = γ 2 and proved that this Diophantine equation has a unique solution in non-negative integers.…”

Section: Introductionmentioning

confidence: 99%

Solution of the Diophantine Equation

Zhou

2023

IRJMETS

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Diophantine equations have several applications in Algebra, Trigonometry; Astronomy; Chemistry; Astrology and Computer Science. These equations play the critical role in solving the problems of balancing the chemical reactions, word problems of algebra and problem of proving the irrationality of a number. In this manuscript, authors studied the Diophantine equation 143 x + 485 y = z 2 , where x, y, z are non-negative integers, and proved that (x, y, z) = (1, 0, 12) is the unique non-negative integer solution of this Diophantine equation.

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Section: Introductionmentioning

confidence: 99%

Solution of the Diophantine Equation

Zhou

2023

IRJMETS

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“…A Diophantine equation with unknowns in exponents is known as an exponential Diophantine equation (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11) . Simultaneous exponential Diophantine equations are a pair of exponential Diophantine equations whose values satisfy both or all of the equations in the collection at the same time.…”

Section: Introductionmentioning

confidence: 99%

A Paradigm for Two Classes of Simultaneous Exponential Diophantine Equations

Pandichelvi,

Vanaja

2023

IJST

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Objectives:The goal of this article is to find integer solutions to two distinct kinds of simultaneous exponential Diophantine equations in three variables. Methods: The system of exponential Diophantine equations is translated into the eminent form of Thue equations, and then their generalised solutions satisfying certain conditions are applied. Findings: The finite set of integer solutions for two disparate categories of simultaneous exponential Diophantine equations consisting of three unknowns is scrutinized. In some circumstances, there is no solution in this analysis for both the dissimilar simultaneous Diophantine equations. Novelty: The motivation is considered to be two types of simultaneous exponential Diophantine equations are first converted into specific system of Pell equations, then into Thue equations for the possibilities of the sum of the exponents, such as x + y = 1 or x + y = 2 . Ifx + y > 3, then, the equations are transformed into a cubic equation, which is not in the form of Pell equations. So, such cases are discarded for exploring solutions to the necessary equations.

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“…Previous works of researchers involve solving the exponential Diophantine equations alone or construct a geometrical shape with some special conditions through Diophantine equations such as Pythagorean equations, Pell equations (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) . This work connects both the ideas and it is undertaken to move this type of research to a somewhat different extent.…”

Section: Introductionmentioning

confidence: 99%

Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers

Mahalakshmi,

Kannan,

Deepshika

et al. 2023

IJST

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Objectives:The specified problem addressed here is the existence and nonexistence of Exponential Diophantine triangles over triangular numbers (t n , n ∈ N). Methods: An Exponential Diophantine triangle over triangular numbers (t n , n ∈ N) is defined as a triangle with sides nx + 1, ny + 2 and nz where x, y, and z are non -negative integers such that t x n + t y n+1 = z 2 . To prove the existence of such triangles, negative Pell's equation and its solutions are used along with some basic number theoretic concepts. To verify the non-existence, the well-known Catalan's conjecture, binomial expansion, and various theories concerning congruence are employed. Findings: Here it is proved that, for five different choices of sides, an Exponential Diophantine triangle over t n can be constructed. In particular, infinitely many such triangles can be found. For some particular choice of sides, Python coding is displayed along with its output to verify the existence of required triangles. On the other side, another five different choices of sides are considered and it is shown that no considered type of triangles exists in these cases. Novelty: The idea of solving an exponential Diophantine equation and the idea of constructing triangles under some conditions using Diophantine equations already exists in the mathematical society. This article is created uniquely by combining these two concepts along with the innovative usage of exponential Diophantine equations.

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On the Exponential Diophantine Equation (13<sup>2m</sup>) + (6r + 1)<sup>n</sup> = z<sup>2</sup>

Cited by 5 publications

References 5 publications

Solution of the Diophantine Equation

Solution of the Diophantine Equation

A Paradigm for Two Classes of Simultaneous Exponential Diophantine Equations

Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers

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