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Relevant bibliographies by topics / Pythagorean quadruples

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Academic literature on the topic 'Pythagorean quadruples'

Author: Grafiati

Published: 2 June 2025

Last updated: 13 July 2025

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Journal articles on the topic "Pythagorean quadruples"

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 makes 4ab a whole square, then (a+b), (a-b) and under root of 4ab turns Pythagorean’s triple. Similarly, utilizing such algebraic identities, Pythagorean’s quadruple up to n-tuples can be generated. If (a+b) is squared, it provides a squared plus b squared plus 2ab. If quantity 2ab is transformed to a whole square on account of assigning values to a or b, then Pythagorean’s quadruples are obtained. Results: Assigning specific values to the terms of simple algebraic identities results in the generation of Pythagorean triples and n-tuples.Conclusions: This paper presents empirical research in which algebraic identities are utilized, resulting in the self-generation of Pythagorean n-tuples. Specific formulas need not be applied, as basic algebraic identities are well known to scholars and students alike. Keywords: Pythagorean’s triples, Quadruples, Quintuples

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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 two squares being odd reduces the search for semi-prime factorization by half. In this paper, we prove that if a Pythagorean quadruple is known and one of its squares represents a Pythagorean triple, then the semi-prime is factorized. The problem of semi-prime factorization is reduced to the problem of finding only one such sum of three squares to factorize a semi-prime. We modify the Lebesgue identity as the sum of four squares to obtain four sums of three squares. These are then expressed as four Pythagorean quadruples. The Brahmagupta–Fibonacci identity reduces these four Pythagorean quadruples to two Pythagorean triples. The greatest common divisors of the sides contained therein are the factors of the semi-prime. We then prove that to factor a semi-prime, it is sufficient that only one of these Pythagorean quadruples be known. We provide the algorithm of our proposed semi-prime factorization method, highlighting its complexity and comparative advantage of the solution space with Fermat’s method. Our algorithm has the advantage when the factors of a semi-prime are congruent to 1 modulus 4. Illustrations of our method for real-world applications, such as factorization of the 768-bit number RSA-768, are established. Further, the computational viabilities, despite the mathematical constraints and the unexplored properties, are suggested as opportunities for future research.

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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 especially for time bound work, since the Diophantine equations are having more than unknown variables.

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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 Yellowstone permutation sequence can generate both primitive and non-primitive Pythagorean triples and quadruples. It demonstrates that the expressions derived from terms in this sequence consistently yield these triples and quadruples through a combination of algebraic properties and geometric interpretations. Additionally, the study formulates propositions to clarify the relationships between the terms of the Yellowstone permutation sequence and their behavior, particularly in generating Pythagorean constructs. The findings underscore the sequence's intriguing mathematical characteristics, offering insights into number theory and its potential applications. This work highlights the role of such sequences in exploring deeper mathematical relationships and fostering curiosity in combinatorial number theory.

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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 + c2 = d2 and then develop an algebraic representation of this method.

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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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Abstract:

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 + c2 = d2 and then develop an algebraic representation of this method.

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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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More sources

Book chapters on the topic "Pythagorean quadruples"

1

Hirayama, Hiroshi. "Numerical Calculation by Quadruple Precision Higher Order Taylor Series Method of the Pythagorean Problem of Three Bodies." In Integral Methods in Science and Engineering. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16077-7_14.

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