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Journal articles on the topic 'Diophantine equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

G, Janaki, and Gowri Shankari A. "(Exponential Diophantine Equation n2􀀀1 )u +n2v = w2;n = 2;3;4;5." Indian Journal of Science and Technology 17, no. 2 (2024): 166–70. https://doi.org/10.17485/IJST/v17i2.2544.

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Abstract <strong>Objectives:</strong> Diophantine research focuses on various ways to tackle multivariable and multidegree Diophantine problems. A Diophantine equation is a polynomial equation with only integer solutions. The objective of this manuscript is to find the solutions to a few exponential Diophantine equations and . Also generalize the Exponential equation , and of the form and explore that it has at least one solution as .<strong> Methods:</strong> Diophantine equations may have finite, infinite or no solutions in integers. There is no universal method for finding so

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2

Sankari, Hasan, and Mohammad Abobala. "On The Group of Units Classification In 3-Cyclic and 4-cyclic Refined Rings of Integers And The Proof of Von Shtawzens' Conjectures." International Journal of Neutrosophic Science 21, no. 4 (2023): 146–54. http://dx.doi.org/10.54216/ijns.210414.

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First Von Shtawzen's Diophantine equation is a non-linear Diophantine equation with three variables . This equation has been conjectured that it has a finite number of integer solutions, and this number of solutions is divisible by 6. Second Von Shtawzen's Diophantine equation is a non-linear Diophantine equation with four variables. This equation has been conjectured that it has a finite number of integer solutions, and this number of solutions is divisible by 8. In this paper, we prove that first Von Shtawzen's conjecture is true, where we show that first Von Shtawzen's Diophantine equations

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3

Aggarwal, S., and S. Kumar. "On the Exponential Diophantine Equation (13^2m) + (6r + 1)ⁿ = z²." Journal of Scientific Research 13, no. 3 (2021): 845–49. http://dx.doi.org/10.3329/jsr.v13i3.52611.

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Nowadays, mathematicians are very interested in discovering new and advanced methods for determining the solution of Diophantine equations. Diophantine equations are those equations that have more unknowns than equations. Diophantine equations appear in astronomy, cryptography, abstract algebra, coordinate geometry and trigonometry. Congruence theory plays an important role in finding the solution of some special type Diophantine equations. The absence of any generalized method, which can handle each Diophantine equation, is challenging for researchers. In the present paper, the authors have d

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4

Wu, Yi, and Zheng Ping Zhang. "The Positive Integer Solutions of a Diophantine Equation." Applied Mechanics and Materials 713-715 (January 2015): 1483–86. http://dx.doi.org/10.4028/www.scientific.net/amm.713-715.1483.

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In this paper, we studied the positive integer solutions of a typical Diophantine equation starting from two basic equations including a Diophantine equation and a Pell equation, and we will prove all the positive integer solutions of the typical Diophantine equation.

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5

Sathiyapriya, R., та M. A. Gopalan. "Homogeneous Quadratic Equation with Four Unknowns 𝑥2 + 𝑥𝑦 + 𝑦2 = 𝑧2 + 𝑧𝑤 + 𝑤2". Indian Journal Of Science And Technology 17, № 27 (2024): 2841–47. http://dx.doi.org/10.17485/ijst/v17i27.1710.

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Objectives: Diophantine research focuses on various ways to tackle multi variable and multi-degree Diophantine problems. A Diophantine equation is a polynomial equation with only integer solutions. The objective of this manuscript is to find the solutions to Polynomial Diophantine equation . Methods: Diophantine equations may have finite, infinite, or no solutions in integers. There is no universal method for finding solutions to Diophantine equations. Different choice of solutions in integers is obtained through using linear transformations and employing the factorization method. Findings: Ma

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6

R, Sathiyapriya, та A. Gopalan M. "Homogeneous Quadratic Equation with Four Unknowns 𝑥2 + 𝑥𝑦 + 𝑦2 = 𝑧2 + 𝑧𝑤 + 𝑤2". Indian Journal of Science and Technology 17, № 27 (2024): 2841–47. https://doi.org/10.17485/IJST/v17i27.1710.

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Abstract <strong>Objectives:</strong> Diophantine research focuses on various ways to tackle multi variable and multi-degree Diophantine problems. A Diophantine equation is a polynomial equation with only integer solutions. The objective of this manuscript is to find the solutions to Polynomial Diophantine equation .<strong> Methods:</strong> Diophantine equations may have finite, infinite, or no solutions in integers. There is no universal method for finding solutions to Diophantine equations. Different choice of solutions in integers is obtained through using linear transforma

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7

Kaleeswari, K., J. Kannan, A. Deepshika, and M. Mahalakshmi. "Computations of Exponential Diophantine Rectangles over Gnomonic Numbers using Python." Indian Journal Of Science And Technology 17, no. 42 (2024): 4449–53. http://dx.doi.org/10.17485/ijst/v17i42.3491.

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Objective: The main objective of this paper is to define and collect a new type of rectangle called the Exponential Diophantine Rectangle over Gnomonic numbers (figurate numbers that take the form 𝑛2 − (𝑛 − 1)2, 𝑛 ∈ 𝑁). Methods: It is done by solving the two exponential Diophantine equations using Mihailescu’s theorem, binomial expansion, and the basic theory of congruences. Findings: Here, it is proven that there are only four exponential Diophantine rectangles over Gnomonic numbers. Finally, it is validated using Python programming for a specific limit. Novelty: The concept of solving an exp

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8

Biswas, D. "Does the Solution to the Non-linear Diophantine Equation 3<sup>x</sup>+35<sup>y</sup>=Z<sup>2</sup> Exist?" Journal of Scientific Research 14, no. 3 (2022): 861–65. http://dx.doi.org/10.3329/jsr.v14i3.58535.

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This paper investigates the solutions (if any) of the Diophantine equation 3x + 35y = Z2, where , x, y, and z are whole numbers. Diophantine equations are drawing the attention of researchers in diversified fields over the years. These are equations that have more unknowns than a number of equations. Diophantine equations are found in cryptography, chemistry, trigonometry, astronomy, and abstract algebra. The absence of any generalized method by which each Diophantine equation can be solved is a challenge for researchers. In the present communication, it is found with the help of congruence th

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9

Chen, Imin. "A Diophantine Equation Associated to X0(5)." LMS Journal of Computation and Mathematics 8 (2005): 116–21. http://dx.doi.org/10.1112/s1461157000000929.

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AbstractSeveral classes of Fermat-type diophantine equations have been successfully resolved using the method of galois representations and modularity. In each case, it is possible to view the proper solutions to the diophantine equation in question as corresponding to suitably defined integral points on a modular curve of level divisible by 2 or 3. Motivated by this point of view, an example of a diophantine equation associated to the modular curve X0(5) is discussed in this paper. The diophantine equation has four terms rather than the usual three terms characteristic of generalized Fermat e

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10

Maran, A. K. "A Simple Solution for Diophantine Equations of Second, Third and Fourth Power." Mapana - Journal of Sciences 4, no. 1 (2005): 96–100. http://dx.doi.org/10.12723/mjs.6.17.

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We know already that the set Of positive integers, which are satisfying the Pythagoras equation Of three variables and four variables cre called Pythagorean triples and quadruples respectively. These cre Diophantine equation OF second power. The all unknowns in this Pythagorean equation have already Seen by mathematicians Euclid and Diophantine. Hcvwever the solution defined by Euclid are Diophantine is also again having unknowns. The only to solve the Diophantine equations wos and error method. Moreover, the trial and error method to obtain these values are not so practical and easy especiall

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11

K, Kaleeswari, Kannan J, Deepshika A, and Mahalakshmi M. "Computations of Exponential Diophantine Rectangles over Gnomonic Numbers using Python." Indian Journal of Science and Technology 17, no. 42 (2024): 4449–53. https://doi.org/10.17485/IJST/v17i42.3491.

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Abstract <strong>Objective:</strong> The main objective of this paper is to define and collect a new type of rectangle called the Exponential Diophantine Rectangle over Gnomonic numbers (figurate numbers that take the form 𝑛2 − (𝑛 − 1)2, 𝑛 ∈ 𝑁). <strong>Methods:</strong> It is done by solving the two exponential Diophantine equations using Mihailescu’s theorem, binomial expansion, and the basic theory of congruences. <strong>Findings:</strong> Here, it is proven that there are only four exponential Diophantine rectangles over Gnomonic numbers. Fi

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12

Biswas, D. "A Discussion on the Solution(s) of the Diophantine Equation 3<sup>x</sup> + 15<sup>y</sup> =Z<sup>2</sup>." Journal of Scientific Research 17, no. 1 (2025): 129–32. https://doi.org/10.3329/jsr.v17i1.74079.

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The absence of a generalized method for solving Diophantine equations having more unknowns than a number of equations is a challenge for researchers in different fields. The presence of the Diophantine equation is reported in the study of the Hydrogen spectrum, quantum Hall effect, chemistry, cryptography, etc. Some special types of Diophantine equations could be addressed with the help of Catalan’s conjecture and Congruence theory. The Diophantine equation 3x + 15y =z2 is addressed in this paper to find the solution(s) in positive integers. It is found that the equation has only two solutions

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13

Biswas, Dibyendu. "A Note on the Diophantine Equation 3x + 63y = z2." Journal of Physical Sciences 29, no. 00 (2024): 23–27. https://doi.org/10.62424/jps.2024.29.00.03.

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Diophantine equations are gradually drawing attention in the study of hydrogen spectrum, eco- nomics, Biology, quantum Hall effect, chemistry, cryptography etc. Different types of schemes are employed to find solution of Diophantine equations. Some special types of Diophantine equations could be addressed with the help of Catalan’s conjecture and Congruence theory. The Diophantine equation (3x+63y=z2) is addressed in this paper to find the solution(s) in non-negative integers. It is found that the equation has only two solutions of (x,y,z) as (1,0,2) and (0,1,8) in non-negative integers.

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14

Aggarwal, Sudhanshu. "On the exponential Diophantine equation \((2^{2m+1}-1)+(13)^n=z^2\)." Engineering and Applied Science Letters 4, no. 1 (2021): 77–79. https://doi.org/10.30538/psrp-easl2021.0064.

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Nowadays, scholars are very interested to determine the solution of different Diophantine equations because these equations have numerous applications in the field of coordinate geometry, cryptography, trigonometry and applied algebra. These equations help us for finding the integer solution of famous Pythagoras theorem and Pell's equation. Finding the solution of Diophantine equations have many challenges for scholars due to absence of generalize methods. In the present paper, author studied the exponential Diophantine equation \((2^{2m+1}-1)+(13)^n=z^2\), where \(m,n\) are whole numbers, for

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15

Song, Junyan. "On Diophantine equation xb=Dy." Theoretical and Natural Science 13, no. 1 (2023): 232–36. http://dx.doi.org/10.54254/2753-8818/13/20240852.

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Diophantine equation is an important part of the number theory and it has been widely studied for a long time. There are many studies on solving the integral solutions of Diophantine equations using algebraic methods. This paper uses documentation method, sums up the results of research in different documents of finding the integral solutions of some Diophantine equations in various conditions, especially the utilization of theories on Pell equation, which in the form of xb=Dy. This paper mainly considers the situations when a =2 or a=3, which is a particular type of Diophantine equation. Seve

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16

Dr., D. Ramprasad. "ON LINEAR DIOPHANTINE EQUATION." International Journal of Applied and Advanced Scientific Research 2, no. 2 (2017): 80–81. https://doi.org/10.5281/zenodo.846868.

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17

V, Pandichelvi, and Saranya S. "Derivation of Finite Number of Integer Solutions to Particular Form of Mordell's Equation a2 = b3+r2; r = 8; 9; 10." Indian Journal of Science and Technology 16, no. 28 (2023): 2113–17. https://doi.org/10.17485/IJST/v16i28.1322.

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Abstract <strong>Objectives:</strong> The perception of solving Diophantine equations is a massive area of research. Mordell equation is a type of Diophantine equation such that the difference of square and cube of numbers remains constant. Various authors analyses Mordell kind equations for existence of solutions by applying several methods. The objective of this manuscript is whether an equation of the form such that the difference of square and cube of numbers provides square of a particular integer. The possibility of visualization of the surface of the considered equation by using th

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18

Dr., R. Sivaraman. "ON SOLVING THREE INTERESTING PUZZLES." International Journal of Scientific Research and Modern Education (IJSRME) 7, no. 1 (2022): 27–29. https://doi.org/10.5281/zenodo.5970145.

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The study of equations whose solutions were in integers was studied for several centuries. Traditionally such equations are known as Diophantine Equations. In this paper, I will pose three related puzzles whose solutions depend on solving a particular Diophantine equation. By providing method for solving that particular equation using continued fraction, I will solve all three puzzles simultaneously.

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19

DĄBROWSKI, ANDRZEJ. "ON A CLASS OF GENERALIZED FERMAT EQUATIONS." Bulletin of the Australian Mathematical Society 82, no. 3 (2010): 505–10. http://dx.doi.org/10.1017/s000497271000033x.

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AbstractWe generalize the main result of the paper by Bennett and Mulholland [‘On the diophantine equation xn+yn=2αpz2’, C. R. Math. Acad. Sci. Soc. R. Can.28 (2006), 6–11] concerning the solubility of the diophantine equation xn+yn=2αpz2. We also demonstrate, by way of examples, that questions about solubility of a class of diophantine equations of type (3,3,p) or (4,2,p) can be reduced, in certain cases, to studying several equations of the type (p,p,2).

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20

Devi, P. Jamuna, та K. S. Araththi. "On the Ternary Cubic Diophantine Equation 𝑥3 + 𝑦3 = 2(𝑧 + 𝑤)2(𝑧 − 𝑤)". Indian Journal Of Science And Technology 17, № 33 (2024): 3473–80. http://dx.doi.org/10.17485/ijst/v17i33.2186.

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The theory of Diophantine equation offers a rich variety of fascinating problems. There are Diophantine problems, which involve cubic equations with four variables. The cubic Diophantine equation given by is analyzed for its patterns of non-zero distinct integral solutions. Objectives: The objective of this paper is to explore the integral solutions of cubic equation by using suitable methodologies. A few interesting relations between the solutions and special numbers are exhibited. Method: Solving Diophantine equation is obtained by the method of Decomposition. The structure of decomposition:

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21

Tatli, Lara, and Paul Stevenson. "A Quantum Diophantine Equation Solution Finder." Mathematical Problems of Computer Science 63 (June 1, 2025): 60–70. https://doi.org/10.51408/1963-0132.

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Diophantine equations are multivariate equations, usually polynomial, in which only integer solutions are admitted. A brute force method for finding solutions would be to systematically substitute possible integer values for the unknown variables and check for equality. Grover’s algorithm is a quantum search algorithm which can find marked indices in a list very efficiently. By treating the indices as the integer variables in the Diophantine equation, Grover’s algorithm can be used to find solutions in a brute force way more efficiently than classical methods. We present a hand-coded example f

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22

DAHMEN, SANDER R. "A REFINED MODULAR APPROACH TO THE DIOPHANTINE EQUATION x2 + y2n = z3." International Journal of Number Theory 07, no. 05 (2011): 1303–16. http://dx.doi.org/10.1142/s1793042111004472.

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Let n be a positive integer and consider the Diophantine equation of generalized Fermat type x2 + y2n = z3 in nonzero coprime integer unknowns x,y,z. Using methods of modular forms and Galois representations for approaching Diophantine equations, we show that for n ∈ {5,31} there are no solutions to this equation. Combining this with previously known results, this allows a complete description of all solutions to the Diophantine equation above for n ≤ 107. Finally, we show that there are also no solutions for n ≡ -1 (mod 6).

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23

Vipawadee, Moonchaisook. "On the Solutions of Diophantine Equation (Mp − 2) x + (Mp + 2) y = z 2 where Mp is Mersenne Prime." International Journal of Basic Sciences and Applied Computing (IJBSAC) 3, no. 4 (2021): 1–3. https://doi.org/10.35940/ijbsac.D0216.083421.

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The Diophantine equation has been studied by many researchers in number theory because it helps in solving variety of complicated puzzle problems. From several studies, many interesting proofs have been found. In this paper, the researcher has examined the solutions of Diophantine equation (𝑴𝒑 − 𝟐) 𝒙 + (𝑴𝒑 + 𝟐) 𝒚 = 𝒛 𝟐 where 𝑴𝒑 is a Mersenne Prime and p is an odd prime whereas x, y and z are nonnegative integers. It was found that this Diophantine equation has no solution. 

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24

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract <strong>Objectives:</strong> The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ). <strong>Methods:</strong> An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . 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 exp

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25

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract <strong>Objectives:</strong> The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ). <strong>Methods:</strong> An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . 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 exp

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26

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract <strong>Objectives:</strong> The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ). <strong>Methods:</strong> An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . 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 exp

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27

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract <strong>Objectives:</strong> The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ). <strong>Methods:</strong> An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . 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 exp

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28

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract <strong>Objectives:</strong> The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ). <strong>Methods:</strong> An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . 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 exp

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29

Ladzoryshyn, N. B., V. M. Petrychkovych, and H. V. Zelisko. "Matrix Diophantine equations over quadratic rings and their solutions." Carpathian Mathematical Publications 12, no. 2 (2020): 368–75. http://dx.doi.org/10.15330/cmp.12.2.368-375.

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The method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equation of same type with triangle coefficients. Solving this matrix equation is reduced to solving a system of linear equations that contains linear Diophantine equations with two variables, their solution methods are well-known. The structure of solutions of matrix equations is also investigated. In

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30

Wadhawan, Narinder Kumar. "SOLUTION TO EQUAL SUM OF FIFTH POWER DIOPHANTINE EQUATIONS - A NEW APPROACH." jnanabha 53, no. 01 (2023): 125–45. http://dx.doi.org/10.58250/jnanabha.2023.53116.

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Purpose of writing this paper is to introduce simple parametric solutions to quintic Diophantine equations 5.n.m where integer n > 2 and integer m > 3. Methodology applied is writing numbers in algebraic form as aI x + bI with variable x, then writing fifth power Diophantine equation, in algebraic form with one variable and then transforming it to a linear equation by vanishing its four terms. For achieving this purpose, values to aI and bI of algebraic numbers are assigned so as to vanish constant term and coefficient of fifth power of x. Then equating with zero the coefficient of secon

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Das, Radhika, Manju Somanath, and V. A. Bindu. "INTEGER SOLUTION ANALYSIS FOR A DIOPHANTINE EQUATION WITH EXPONENTIALS." jnanabha 53, no. 02 (2023): 69–73. http://dx.doi.org/10.58250/jnanabha.2023.53208.

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The exponential Diophantine equation is one of the distinctive types of Diophantine equations where the variables are expressed as exponents. For these equations, considerable excellent research has already been done. In this study, we try to solve the equations 3λ + 103μ = ξ2, 3λ + 181μ = ξ2, 3λ + 193μ = ξ2.

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32

Mude, Lao Hussein, Kinyanjui Jeremiah Ndung’u, and Zachary Kaunda Kayiita. "On Sums of Squares Involving Integer Sequence: \(\sum_{r=1}^n w_r^2+\frac{n}{3} d^2=3\left(\frac{n d^2}{3}+\sum_{r=1}^{\frac{n}{3}} w_{3 r-1}^2\right)\)." Journal of Advances in Mathematics and Computer Science 39, no. 7 (2024): 1–6. http://dx.doi.org/10.9734/jamcs/2024/v39i71906.

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Let wr be a given integer sequence in arithmetic progression with a common difference d. The study of diophantine equations, which are polynomial equations seeking integer solutions, has been a very interesting journey in the field of number theory. Historically, these equations have attracted the attention of many mathematicians due to their intrinsic challenges and their significance in understanding the properties of integers. In this current study, we examine a diophantine equation relating the sum of squared integers from specific sequences to a variable d: In particular, the diophantine

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33

Mangalaeng, Aswad Hariri. "The Primitive-Solutions of Diophantine Equation x^2+pqy^2=z^2, for primes p,q." Jurnal Matematika, Statistika dan Komputasi 18, no. 2 (2022): 308–14. http://dx.doi.org/10.20956/j.v18i2.19018.

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In this paper, we determine the primitive solutions of diophantine equations x^2+pqy^2=z^2, for positive integers x, y, z, and primes p,q. This work is based on the development of the previous results, namely using the solutions of the Diophantine equation x^2+y^2=z^2, and looking at characteristics of the solutions of the Diophantine equation x^2+3y^2=z^2 and x^2+9y^2=z^2.

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Tyszka, Apoloniusz. "A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions." Open Computer Science 8, no. 1 (2018): 109–14. http://dx.doi.org/10.1515/comp-2018-0012.

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Abstract We define a computable function f from positive integers to positive integers. We formulate a hypothesis which states that if a system S of equations of the forms xi· xj = xk and xi + 1 = xi has only finitely many solutions in non-negative integers x1, . . . , xi, then the solutions of S are bounded from above by f (2n). We prove the following: (1) the hypothesis implies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if

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P, Jamuna Devi, та S. Araththi K. "On the Ternary Cubic Diophantine Equation 𝑥3 + 𝑦3 = 2(𝑧 + 𝑤)2(𝑧 − 𝑤)". Indian Journal of Science and Technology 17, № 33 (2024): 3473–80. https://doi.org/10.17485/IJST/v17i33.2186.

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Abstract The theory of Diophantine equation offers a rich variety of fascinating problems. There are Diophantine problems, which involve cubic equations with four variables. The cubic Diophantine equation given by is analyzed for its patterns of non-zero distinct integral solutions. <strong>Objectives:</strong> The objective of this paper is to explore the integral solutions of cubic equation by using suitable methodologies. A few interesting relations between the solutions and special numbers are exhibited. <strong>Method:</strong> Solving Diophantine equation is obtained

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Abobala, Mohammad. "A Short Contribution to Split-Complex Linear Diophantine Equations in Two Variables." Galoitica: Journal of Mathematical Structures and Applications 6, no. 2 (2023): 32–35. http://dx.doi.org/10.54216/gjmsa.060204.

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In this work, we study the split-complex integer solutions for the split-complex linear Diophantine equation in two variables where are split-complex integers. An algorithm for generating all solutions will be obtained by transforming the split-complex equation to a classical equivalent system of linear Diophantine equations in four variables.

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37

Cassels, J. W. S. "A diophantine equation." Glasgow Mathematical Journal 27 (October 1985): 11–18. http://dx.doi.org/10.1017/s0017089500006030.

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I was recently challenged to find all the cases when the sum of three consecutive integral cubes is a square; that is to find all integral solutions x, y ofy2=(x−1)3+x3+(x+1)3=3x(x2+2)This is an example of a curve of genus 1. There is an effective procedure for finding all integral points on a given curve of genus 1 ([1, Theorem 4.2], [2]): that is, it can be guaranteed to find all the integral points and to show that no others exist with a finite amount of work. Unlike some effective procedures, which have only logical interest, this one can actually be carried out in practice, at least with

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38

Obiero, Beatrice Adhiambo, and Kimtai Boaz Simatwo. "On Certain Results On The Diophantine Equation: \(\sum_{{r}={1}}^{n}w^2_r+\frac{n}{3}d^2=3(\frac{nd^2}{3}+\sum^{\frac{n}{3}}_{r=1}w^2_{3r-1})\)." Journal of Advances in Mathematics and Computer Science 40, no. 2 (2025): 1–7. https://doi.org/10.9734/jamcs/2025/v40i21966.

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Consider a sequence wr in arithmetic progression with a common difference d. he exploration of Diophantine equations, which are polynomial equations seeking integer solutions, has been a fascinating endeavor in number theory. These equations have historically intrigued mathematicians due to their inherent complexities and their importance in understanding the properties of integers. In this study, we investigate a Diophantine equation that relates the sum of squares of integers from specific sequences to a variable d. Specifically, we extend existing results on the Diophantine equation: \(\sum

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Alfahal, Abuobida M. A., Yaser A. Alhasan, Raja A. Abdulfatah, and Rozina Ali. "On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers." International Journal of Neutrosophic Science 20, no. 3 (2023): 08–14. http://dx.doi.org/10.54216/ijns.200301.

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The Diophantine equation X^n+Y^n=Z^n is called the Fermat's Diophantine equation. Its solutions are called general Fermat's triples.The aim of this paper is to study the solutions of Fermat's Diophantine equation in the 2-cyclic refined neutrosophic ring of integers, where we determine all possible solutions of this Diophantine equation, as well as, the special case of Pythagoras triples.

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Dr., R. Sivaraman. "ON SOLVING A QUADRATIC DIOPHANTINE EQUATION." Deutsche internationale Zeitschrift für zeitgenössische Wissenschaft 43 (November 2, 2022): 19–20. https://doi.org/10.5281/zenodo.7273715.

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Diophantine Equations named after ancient Greek mathematician Diophantus, plays a vital role not only in number theory but also in several branches of science. In this paper, we will solve one of the quadratic Diophantine equations and provide its complete solutions. The method adopted to solve the given equation is using the concept of polar form of a particular complex number. This concept can be generalized for solving similar equations.

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A’lailliyyin1, A’lailliyyin, Rica Amalia, and Tony Yulianto. "Penerapan Kombinatorik pada Persamaan Diophantine Linier." Zeta - Math Journal 7, no. 2 (2022): 64–68. http://dx.doi.org/10.31102/zeta.2022.7.2.64-68.

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In this paper we apply combinatoric principle on linear diophantine equation. The method being used on this research is theoritic method. Generally, the linear diophantine equation is a polynomial equation. The form of equation being solved in this paper is linear diophantine equation. As for the solution of this equation is limited to natural numbers and whole numbers. By using the combinatoric principle, we got the number of solution of these linear diophantine equation which is the combination for natural numbers solution and for whole numbers solution.

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Ossicini, Andrea. "On the Nature of Some Euler’s Double Equations Equivalent to Fermat’s Last Theorem." Mathematics 10, no. 23 (2022): 4471. http://dx.doi.org/10.3390/math10234471.

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In this work, I provide a new rephrasing of Fermat’s Last Theorem, based on an earlier work by Euler on the ternary quadratic forms. Effectively, Fermat’s Last Theorem can be derived from an appropriate use of the concordant forms of Euler and from an equivalent ternary quadratic homogeneous Diophantine equation able to accommodate a solution of Fermat’s extraordinary equation. Following a similar and almost identical approach to that of A. Wiles, I tried to translate the link between Euler’s double equations (concordant/discordant forms) and Fermat’s Last Theorem into a possible reformulation

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43

Tadee, Suton, and Apirat Siraworakun. "Nonexistence of Positive Integer Solutions of the Diophantine Equation p^x + (p + 2q)^ y = z^2 , where p, q and p + 2q are Prime Numbers." European Journal of Pure and Applied Mathematics 16, no. 2 (2023): 724–35. http://dx.doi.org/10.29020/nybg.ejpam.v16i2.4702.

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The Diophantine equation p^x + (p + 2q)^y = z^2 , where p, q and p + 2q are prime numbers, is studied widely. Many authors give q as an explicit prime number and investigate the positive integer solutions and some conditions for non-existence of positive integer solutions. In this work, we gather some conditions for odd prime numbers p and q for showing that the Diophantine equation p^x + (p + 2q)^y = z^2 has no positive integer solution. Moreover, many examples of Diophantine equations with no positive integer solution are illustrated.

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Jin, Yuan, and Asmus L. Schmidt. "A Diophantine equation appearing in Diophantine approximation." Indagationes Mathematicae 12, no. 4 (2001): 477–82. http://dx.doi.org/10.1016/s0019-3577(01)80036-7.

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Tahiliani, Dr Sanjay. "More on Diophantine Equations." International Journal of Management and Humanities 5, no. 6 (2021): 26–27. http://dx.doi.org/10.35940/ijmh.l1081.025621.

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In this paper, we will find the solutions of many Diophantine equations.Some are of the form 2(3 x )+ 5(7y ) +11=z2 for non negative x,y and z. we also investigate solutions ofthe Diophantine equation 2(x+3) +11(3y ) ─ 1= z2 for non negative x,y and z and finally, westudy the Diophantine equations (k!×k)n = (n!×n)k and ( 𝒌! 𝒌 ) 𝒏 = ( 𝒏! 𝒏 ) 𝒌 where k and n are positive integers. We show that the first one holds if and only if k=n and the second one holds if and only if k=n or (k,n) =(1,2),(2,1).We also investigate Diophantine equation u! + v! = uv and u! ─ v! = uv .

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Xiong, Ming. "Solving linear Diophantine equation and simultaneous linear Diophantine equations with minimum principles." International Mathematical Forum 17, no. 4 (2022): 143–61. http://dx.doi.org/10.12988/imf.2022.912318.

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M., Vasuki, and R. Sivaraman Dr. "SOLVING QUADRATIC DIOPHANTINE EQUATION FOR INTEGRAL POWERS OF 37." INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH 12, no. 01 (2024): 3996–98. https://doi.org/10.5281/zenodo.10604267.

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Diophantine Equations named after ancient Greek mathematician Diophantus, plays a vital role not only in number theory but also in several branches of science. In this paper, we have solved an quadratic Diophantine equations where the right hand side are positive integral powers of 37 and provide its integer solutions. The method adopted to solve the given equation is using the concept of polar form of a particular complex number. This concept can be generalized for solving similar equations.

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Peszek, Agnieszka, and Apoloniusz Tyszka. "On the Relationship Between Matiyasevich's and Smoryński's Theorems." Scientific Annals of Computer Science XXIX, no. 1 (2019): 101–11. https://doi.org/10.7561/SACS.2019.1.101.

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Let R be a non-zero subring of Q with or without 1. We assume that for every positive integer n there exists a computable surjection from N onto R<sup>n</sup> . Every R ∈ {Z, Q} satisfies these conditions. Matiyasevich’s theorem states that there is no algorithm to decide whether or not a given Diophantine equation has a solution in non-negative integers. Smoryński’s theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. We prove: (1) Smoryński’s theorem easily follows fr

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Wu, Yi, Lan Long та Zheng Ping Zhang. "The Integer Solutions of Diophantine Equation Χ²− 21 = 4y⁵". Applied Mechanics and Materials 687-691 (листопад 2014): 1182–85. http://dx.doi.org/10.4028/www.scientific.net/amm.687-691.1182.

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In this paper, we studied the integer solutions of the typical Diophantine equations with some important theories in quadratic fields and the fundamental theorem of arithmetic in the ring of quadratic algebraic integers. We proved all the integer solutions of the Diophantine equation Χ2 − 21 = 4y5.

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Tadee, Suton. "On the Diophantine Equation 1/w+1/x+1/y+1/z=u/u+1." Progress in Applied Science and Technology 15, no. 1 (2025): 1–5. https://doi.org/10.60101/past.2025.257129.

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In 2023, Wongsanurak and Duangdai found all positive integer solutions of the Diophantine equation , when and are positive integers with and . In this work, by using an elementary approach, we solved the Diophantine equation for any positive integer and . The results of the research found that the Diophantine equation under the above conditions has twenty-seven positive integer solutions.

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