Relevant bibliographies by topics / Pythagorean n-tuples

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Academic literature on the topic 'Pythagorean n-tuples'

Author: Grafiati
Published: 2 June 2025
Last updated: 19 July 2025

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Journal articles on the topic "Pythagorean n-tuples"

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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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Cass, Daniel, and Pasquale J. Arpaia. "Matrix Generation of Pythagorean n-Tuples." Proceedings of the American Mathematical Society 109, no. 1 (1990): 1. http://dx.doi.org/10.2307/2048355.

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Cass, Daniel, and Pasquale J. Arpaia. "Matrix generation of Pythagorean $n$-tuples." Proceedings of the American Mathematical Society 109, no. 1 (1990): 1. http://dx.doi.org/10.1090/s0002-9939-1990-1000148-0.

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4

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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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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Song, Haizhou, and Wang Qiufen. "Property and Representation of n-Order Pythagorean Matrix." Mathematical Problems in Engineering 2020 (March 24, 2020): 1–10. http://dx.doi.org/10.1155/2020/2857417.

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Here we study the character and expression of n-order Pythagorean matrix using number theory. Theories of Pythagorean matrix are obtained. Using related algebra skills, we prove that the set which constitutes all n-order Pythagorean matrices is a finitely generated group of matrix multiplication and gives a generated tuple of this finitely generated group (n≤10) simultaneously.
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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 formula characterizes an umbilical round hypersphere satisfying k1=k2=⋯=kn, i.e., Hn=Ke in Mn+1(c).
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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. Moreover, in the set of Pythagorean quadruples, we aim to find a commutative in8nite group with elements in Q or Z. To achieve these results, we prove the existence of some suitable binary operations.
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Book chapters on the topic "Pythagorean n-tuples"

1

Thiruchinapalli, Srinivas, and C. Ashok Kumar. "Construction of Pythagorean and Reciprocal Pythagorean n-tuples." In Springer Proceedings in Mathematics & Statistics. Springer Nature Switzerland, 2024. http://dx.doi.org/10.1007/978-3-031-51163-9_4.

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