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Journal articles on the topic 'Pythagorean quadruples'

Author: Grafiati

Published: 2 June 2025

Last updated: 31 July 2025

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1

Wadhawan, Narinder. "SELF GENERATING n-TUPLES." Graduate Journal of Interdisciplinary Research, Reports and Reviews 1, no. 1 (2023): 18–27. https://doi.org/10.34256/gjir3.v1i1.5.

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Background: The Pythagorean triple based on Pythagorean Theorem, were known in to ancient Babylon and Egypt. The interrelation of the three was known as far back as thousands of years, but it was Pythagoras who explicitly explained their equation.Purpose: Different methods have been put forth by the mathematicians for generation of Pythagorean’s triple and n-tuples but this paper provides a unique method how these get self-generated by use of simple algebraic expansions.Methods: An algebraic quantity (a+b) squared equals to (a-b) squared plus 4ab and if a or b is assigned such a value that mak

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2

Overmars, Anthony, and Sitalakshmi Venkatraman. "New Semi-Prime Factorization and Application in Large RSA Key Attacks." Journal of Cybersecurity and Privacy 1, no. 4 (2021): 660–74. http://dx.doi.org/10.3390/jcp1040033.

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Semi-prime factorization is an increasingly important number theoretic problem, since it is computationally intractable. Further, this property has been applied in public-key cryptography, such as the Rivest–Shamir–Adleman (RSA) encryption systems for secure digital communications. Hence, alternate approaches to solve the semi-prime factorization problem are proposed. Recently, Pythagorean tuples to factor semi-primes have been explored to consider Fermat’s Christmas theorem, with the two squares having opposite parity. This paper is motivated by the property that the integer separating these

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3

Nelsen, Roger. "Proof Without Words: Pythagorean Quadruples." College Mathematics Journal 45, no. 3 (2014): 179. http://dx.doi.org/10.4169/college.math.j.45.3.179.

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4

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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5

Ojera, Dariel. "Unveiling the Properties and Relationship of Yellowstone Permutation Sequence." Psychology and Education: A Multidisciplinary Journal 27, no. 2 (2024): 173–84. https://doi.org/10.5281/zenodo.13993059.

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This paper explores a mathematical sequence known as the Yellowstone permutation, introduced by Zumkeller (2004). This sequence, characterized by alternating even and odd integers with prime and composite number patterns, is studied for its unique properties and connections to mathematical structures like Pythagorean triples and quadruples. The research employs descriptive and expository methods to explore the sequence’s nature, establishing it as infinite, containing infinitely many primes, and ensuring that all integers appear at least once. The paper also delves into how the Yellowsto

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6

Booze, David A. "Delving Deeper: Visualizing Pythagorean Triples and Beyond." Mathematics Teacher 104, no. 5 (2010): 393–98. http://dx.doi.org/10.5951/mt.104.5.0393.

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Exploring the algebraic properties that form the framework for analytical generation of Pythagorean triples and quadruples is a challenging topic for high school algebra students. I have had considerable success in motivating my students to explore these properties by using a visual method. Students can use this visual method to find Pythagorean triples and quadruples easily and can capably and colorfully supply as many integer solutions as they desire to these well-known equations. Here I will present a visual method for producing integer solutions to the equations a2 + b2 = c2 and a2 + b2 +

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7

Booze, David A. "Delving Deeper: Visualizing Pythagorean Triples and Beyond." Mathematics Teacher 104, no. 5 (2010): 393–98. http://dx.doi.org/10.5951/mt.104.5.0393.

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Exploring the algebraic properties that form the framework for analytical generation of Pythagorean triples and quadruples is a challenging topic for high school algebra students. I have had considerable success in motivating my students to explore these properties by using a visual method. Students can use this visual method to find Pythagorean triples and quadruples easily and can capably and colorfully supply as many integer solutions as they desire to these well-known equations. Here I will present a visual method for producing integer solutions to the equations a2 + b2 = c2 and a2 + b2 +

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8

M., Stupel, Sigler Avi(Berman), and Jahangiri J. "A generalization of Pythagorean triples for desirable quadrilaterals." Journal of Progressive Research in Mathematics 13, no. 2 (2018): 2282–90. https://doi.org/10.5281/zenodo.3974664.

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We explore the generalization of famous Pythagorean triples (a, b, c) for triangles to Pythagorean quadruples (a, b, c, d) for desirable quadrilaterals. Using number theory and geometrical techniques including Diophantine equations and Ptolemy’s Theorem, we show that there are infinite number of such quadrilaterals with specific relations between their sides and diagonals. We conclude our paper with an open question for further investigation.

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9

Frisch, Sophie, and Leonid Vaserstein. "Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples." Journal of Pure and Applied Algebra 216, no. 1 (2012): 184–91. http://dx.doi.org/10.1016/j.jpaa.2011.06.002.

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10

Oliverio, Paul. "Self-Generating Pythagorean Quadruples and N -Tuples." Fibonacci Quarterly 34, no. 2 (1996): 98–101. http://dx.doi.org/10.1080/00150517.1996.12429074.

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11

Güler, Erhan, Yusuf Yaylı, and Magdalena Toda. "Differential Geometry and Matrix-Based Generalizations of the Pythagorean Theorem in Space Forms." Mathematics 13, no. 5 (2025): 836. https://doi.org/10.3390/math13050836.

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In this work, we consider Pythagorean triples and quadruples using fundamental form matrices of hypersurfaces in three- and four-dimensional space forms and illustrate various figures. Moreover, we generalize that an immersed hypersphere Mn with radius r in an (n+1)-dimensional Riemannian space form Mn+1(c), where the constant sectional curvature is c∈{−1,0,1}, satisfies the (n+1)-tuple Pythagorean formula Pn+1. Remarkably, as the dimension n→∞ and the fundamental form N→∞, we reveal that the radius of the hypersphere converges to r→12. Finally, we propose that the determinant of the Pn+1 form

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12

Wadhawan, Narinder Kumar, and Priyanka Wadhawan. "A NEW APPROACH TO GENERATE FORMULAE FOR PYTHAGOREANS TRIPLES, QUADRUPLES AND THEIR GENERALISATION TO N-TUPLES." jnanabha 50, no. 02 (2020): 200–211. http://dx.doi.org/10.58250/jnanabha.2020.50224.

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In this paper, innovative methods have been devised to generate formulae for Pythagorean’s Triples, Quadruples and these are finally generalised to generate Pythagorean’s n-tuples. First method utilises formula for solution of a quadratic equation and generate two sets of Pythagorean’s Triples. Second method determines universal identities that satisfy Pythagorean’s Triples, Quadruples so on up to n-tuples. These methods are unprecedented, easy to derive at and hence are comprehensible to students and scholars alike.

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13

Galarza, Fredy Canizares, Marcos Lalama Flores, Diego Palma Rivero, and Mohammad Abobala. "On Weak Fuzzy Complex Pythagoras Quadruples." International Journal of Neutrosophic Science 22, no. 2 (2023): 108–13. http://dx.doi.org/10.54216/ijns.220209.

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In this work, we study the generating of Pythagoras quadruples in the sets of weak fuzzy complex integers and anti-weak fuzzy complex integers, where we present sufficient and necessary conditions for generating Pythagoras quadruples in the mentioned sets. Also, we present many examples to clarify our work's validity and novelty.

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14

Gandhi, K. Raja Rama, and D. Narasimha Murty. "Generalization of Pythagorean Triplets, Quadruple." Bulletin of Society for Mathematical Services and Standards 1 (March 2012): 40–45. http://dx.doi.org/10.18052/www.scipress.com/bsmass.1.40.

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The method of computing Pythagorean triples is well known. All though the classical formulas produce all primitive triples, which do not generate all possible triples, specially non-primitive triples. This paper presents a novel approach to produce all likely triples both primitive and non-primitive, Quadruple for any extent.

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15

Merkepci, Hamiyet, and Ahmed Hatip. "Algorithms for Computing Pythagoras Triples and 4-Tiples in Some Neutrosophic Commutative Rings." International Journal of Neutrosophic Science 20, no. 3 (2023): 107–14. http://dx.doi.org/10.54216/ijns.200310.

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This paper is dedicated to study the number theoretical Pythagoras triples\4-tiples problem in several kinds of neutrosophic algebraic systems, where it finds an algorithm to find Pythagoras triples\4-tiples in commutative neutrosophic rings and refined neutrosophic rings too. Besides, the necessary and sufficient condition for a triple\4-tiple to be Pythagoras triple\4-tiple (quadruples) is obtained and proven in term of theorems. In addition, many numerical examples will be illustrated.

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16

Rashel, Rashel, and Murhaf Obaidi. "On Some Novel Results About Fuzzy n-Standard Number Theoretical Systems and Fuzzy Pythagoras Triples." Journal of Neutrosophic and Fuzzy Systems 8, no. 1 (2024): 18–22. http://dx.doi.org/10.54216/jnfs.080102.

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The main goal of this work is to study the solutions of linear congruencies in the standard and n-standard fuzzy number theoretical systems, where we present necessary and sufficient conditions for fuzzy congruence to be solvable in these systems. Also, we provide the conditions of fuzzy nilpotency and fuzzy invertibility modulo integers with many illustrated examples. On the other hand, we suggest an algorithm to generate fuzzy Pythagoras triples and fuzzy Pythagoras quadruples in the standard fuzzy number theoretical system.

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17

Zargar, Arman Shamsi. "On the rank of elliptic curves arising from Pythagorean quadruplets." Kodai Mathematical Journal 43, no. 1 (2020): 129–42. http://dx.doi.org/10.2996/kmj/1584345690.

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18

Moody, Dustin, and Arman Shamsi Zargar. "On the rank of elliptic curves arising from Pythagorean quadruplets, II." Colloquium Mathematicum 163, no. 2 (2021): 189–96. http://dx.doi.org/10.4064/cm8101-3-2020.

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19

Zambrano, Luis Albarracín, Fabricio Lozada Torres, Bolívar Villalta Jadan, and Nabil Khuder Salman. "On Symbolic 7-Plithogenic and 8-Plithogenic Number Theoretical Concepts." International Journal of Neutrosophic Science 22, no. 4 (2023): 44–55. http://dx.doi.org/10.54216/ijns.220404.

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This paper is dedicated to studying the foundations of 7-plithogenic and 8-plithogenic number theory, where the central concepts about symbolic 7-plithogenic8-plithogenic integers will be discussed such as symbolic 7-plithogenic8-plithogenic Pythagoras triples and quadruples, symbolic 7-plithogenic8-plithogenic linear Diophantine equations, and the divisors. On the other hand, we prove that Euler's theorem is still true in the case of the symbolic 7-plithogenic8-pithogenic number theory.

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20

Świniarski, Janusz. "Philosophy and Social Sciences in a Securitological Perspective." Polish Political Science Yearbook 52 (2022): 1–23. http://dx.doi.org/10.15804/ppsy202302.

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The inspiration of this text is the belief of the Pythagoreans that the roots and source of complete knowledge is the quadruple expressed in the “arch-four”, also called as tetractys. Hence the hypothesis considered in this paper is: the basis of the philosophy of social sciences is entangled in these four valours, manifested in what is “general and necessary” (scientific) in social life, the first and universal as to the “principles and causes” of this life (theoretically philosophical) and “which can be different in it” (practically philosophical) and “intuitive”. The quadruple appears with

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21

AMATO, Roberto. "Characterization of Diophantine Equations a + y^2 = z^2, Pythagorean n-Tuples, and Algebraic Structures." International Journal of Mathematics and Mathematical Sciences 2025 (May 24, 2025). https://doi.org/10.1155/ijmm/5516311.

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Let N, Z, and Q be the sets of natural, integers, and rational numbers, respectively. Our objective, involving a predeterminedpositive integer a, is to study a characterization of Diophantine equations of the form a + y^2 = z^2 . Building on this result, we aim to obtain a characterization for Pythagorean n-tuples. Furthermore, we seek to prove the existence of a commutative infinitemonoid in the set of Diophantine equations a + y^2 = z^2 with elements in N. Additionally, we intend to establish a commutative infinite monoid with elements in N or Z on the set of Pythagorean quadruples. Mor

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22

A., Alfahal Abuobida, Yaser Alhasan, Raja Abdulfatah, and Sara Sawalmeh. "On The Conditions for Symbolic 3-Plithogenic Pythagoras Quadruples." October 22, 2023. https://doi.org/10.5281/zenodo.10031190.

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