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Journal articles on the topic 'Pythagorean triangles'

Author: Grafiati

Published: 3 June 2025

Last updated: 19 July 2025

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1

Mita Darbari and Prashans Darbari. "Pythagorean triangles with sum of its two legs as Dodecic." GSC Advanced Engineering and Technology 3, no. 1 (2022): 011–15. http://dx.doi.org/10.30574/gscaet.2022.3.1.0028.

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Number Theory is almost four thousand years old. In ancient clay tablet in Babylon, integral solutions of Pythagorean equations were listed down. Pythagoras theorem has always fascinated young and old mind alike. More than four hundred proofs of the Pythagoras theorem have been discovered so far. Pythagoras theorem is applied in almost every sphere of science- from geometry to Einstein’s theory of relativity. New Pythagorean triangles are being discovered which satisfy certain constraints. Integral solutions of Diophantine equations related to Pythagorean equation are sought by many mathematic

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2

A., Dinesh Kumar, and Vasuki M. "A STUDY ON PYTHAGOREAN TRIPLES." International Journal of Interdisciplinary Research in Arts and Humanities 1, no. 1 (2016): 14–21. https://doi.org/10.5281/zenodo.155337.

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The Pythagorean numbers play a significant role in the theory of higher arithmetic as they come in the majority of indeterminate problem. For the discovery of the law of the three squares (Pythagorean equation), really, one should be indebted to the Pythagorean who were the first Greeks with great intellectual perception. One may notice to his surprise that the Egyptians, the Chinese, the Babylonians and the Indians knew some knowledge of the property of right angled Pythagorean triangles or Pythagorean numbers .Since there is a 1−1 correspondence between Pythagorean numbers and Pythagorean tr

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3

Shanthi, J., та M. A. Gopalan. "Formulation of Special Pythagorean Triangles through Integer Solutions of the Hyperbola 𝑦𝟐 = (𝑘𝟐 + 𝟐𝑘)𝑥𝟐 + 𝟏". Indian Journal Of Science And Technology 17, № 41 (2024): 4307–12. http://dx.doi.org/10.17485/ijst/v17i41.3132.

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Objectives: The objective of this research paper is to formulate Pythagorean triangles, each of which satisfies the relation: Hypotenuse = (𝑘 + 1) times the leg with even values added with unity, through employing the integer solutions to the Hyperbola 𝑦2 = (𝑘2 + 2𝑘)𝑥2 + 1. Methods: The sides of the Pythagorean triangle satisfying the requirement are obtained by suitably choosing its generators consisting of the integer solutions to the considered hyperbola. Findings : There are plenty of Pythagorean triangles satisfying the given characterization for each value of k > 0 in the binary quadr

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J, Shanthi, та A. Gopalan M. "Formulation of Special Pythagorean Triangles through Integer Solutions of the Hyperbola 𝑦𝟐 = (𝑘𝟐 + 𝟐𝑘)𝑥𝟐 + 𝟏". Indian Journal of Science and Technology 17, № 41 (2024): 4307–12. https://doi.org/10.17485/IJST/v17i41.3132.

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Abstract <strong>Objectives:</strong> The objective of this research paper is to formulate Pythagorean triangles, each of which satisfies the relation: Hypotenuse = (𝑘 + 1) times the leg with even values added with unity, through employing the integer solutions to the Hyperbola 𝑦2 = (𝑘2 + 2𝑘)𝑥2 + 1. <strong>Methods:</strong> The sides of the Pythagorean triangle satisfying the requirement are obtained by suitably choosing its generators consisting of the integer solutions to the considered hyperbola. <strong>Findings :</strong> There are plenty of Pythagorean triangles

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5

Rajput, Chetansing. "Metallic Ratios in Primitive Pythagorean Triples : Metallic Means embedded in Pythagorean Triangles and other Right Triangles." JOURNAL OF ADVANCES IN MATHEMATICS 20 (July 15, 2021): 312–44. http://dx.doi.org/10.24297/jam.v20i.9088.

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The Primitive Pythagorean Triples are found to be the purest expressions of various Metallic Ratios. Each Metallic Mean is epitomized by one particular Pythagorean Triangle. Also, the Right Angled Triangles are found to be more “Metallic” than the Pentagons, Octagons or any other (n2+4)gons. The Primitive Pythagorean Triples, not the regular polygons, are the prototypical forms of all Metallic Means.

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6

Hikmatovich, Ibragimov Husniddin. "Connection Between A Right Triangle And An Equal Side Triangle." American Journal of Interdisciplinary Innovations and Research 02, no. 11 (2020): 105–14. http://dx.doi.org/10.37547/tajiir/volume02issue11-20.

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There is some evidence that a right triangle and an equilateral triangle are related. Information about Pythagorean numbers is given. The geometric meaning of the relationship between right triangles and equilateral triangles is shown. The geometric meaning of the relationship between an equilateral triangle and an equilateral triangle is shown.

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7

Swenson, Carl, and André Yandl. "Partitioning Pythagorean Triangles Using Pythagorean Angles." College Mathematics Journal 43, no. 3 (2012): 220–25. http://dx.doi.org/10.4169/college.math.j.43.3.220.

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8

Choi, Eunmi. "FIBONACCI TYPE PYTHAGOREAN TRIANGLES." Far East Journal of Mathematical Sciences (FJMS) 101, no. 9 (2017): 2023–41. http://dx.doi.org/10.17654/ms101092023.

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9

S., Yahya Mohamed *. &. M. Prem Kumar. "A NOTE ON SPECIAL PAIRS OF PYTHAGOREAN TRIANGLE AND 3-DIGIT SPHENIC NUMBER." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 6, no. 7 (2017): 242–46. https://doi.org/10.5281/zenodo.823122.

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In this paper , we present pairs of Pythagorean triangles such in each pair, the difference between their perimeters is four times the 3-digit Sphenic number 110. Also we present the number of pairs of primitive and non- primitive Pythagorean triangles.

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10

S., Yahya Mohamed *1 S. Prem Kumar 2. and V. Somu3. "SPECIAL PAIRS OF PYTHAGOREAN TRIANGLES AND 3 –DIGITS HARSHAD NUMBER." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 6, no. 8 (2017): 226–29. https://doi.org/10.5281/zenodo.843869.

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Here, we present pairs of Pythagorean triangles such that in each pair, the difference between their perimeters is four times the 3-digit Harshad number 108. Also we present the number of pairs of primitive and non- primitive Pythagorean triangles.

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11

Calzavara, Luigi, and Maurizio Brizzi. "geometry of Venice." Advances in Methodology and Statistics 1, no. 1 (2004): 265–75. http://dx.doi.org/10.51936/havv8635.

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Having observed that ancient Venice belfries are located in such a way that they generate many Pythagorean triangles, having a great number of vertices in common, it has been decided to test the null hypothesis of random location by statistical and probabilistic methods. A simple index, called Pythagorean Ratio, is proposed, for checking which triangles are to be considered as Pythagorean. Then, a Monte Carlo simulation is performed, generating samples of "random belfries" in the historical kernel of Venice; a Poisson model seems to fit very well the number X of Pythagorean triangles. Combinin

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12

Bolognese, Chris A. "Mathematical Lens: How Much Can You Bench?" Mathematics Teacher 106, no. 6 (2013): 414–17. http://dx.doi.org/10.5951/mathteacher.106.6.0414.

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Students analyze a photograph to solve mathematical questions related to the images captured in the photograph. This month the mathematics behind the photograph includes finding areas of regular polygons, right triangle trigonometry, the Pythagorean theorem, special right triangles, and similarity and scale factors.

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13

Koszegyova, Lucia, Evelin Csókási, and Juraj Hirjak. "Structure of Primitive Pythagorean Triples in Generating Trees." European Journal of Pure and Applied Mathematics 17, no. 3 (2024): 2127–41. http://dx.doi.org/10.29020/nybg.ejpam.v17i3.5323.

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A Pythagorean triple is a triple of positive integers $(a,b,c)$ such that $a^2+b^2=c^2$. If $a,b$ are coprime, then it is called a primitive Pythagorean triple. It is known that every primitive Pythagorean triple can be generated from the triple $(3,4,5)$ using multiplication by unique number and order of three specific $3\times3$ matrices, which yields a ternary tree of triplets. Two such trees were described by Berggren and Price, respectively. A different approach is to view the primitive Pythagorean triples as points in the three-dimensional Euclidean space. In this paper, we prove that th

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14

Jepsen, Charles, and Roc Yang. "Making Squares from Pythagorean Triangles." College Mathematics Journal 29, no. 4 (1998): 284. http://dx.doi.org/10.2307/2687683.

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15

Jayakumar, P., and G. Shankarakalidoss. "Hexagonal Numbers and Pythagorean Triangles." Journal of Mathematics and Informatics 11 (December 11, 2017): 13–15. http://dx.doi.org/10.22457/jmi.v11a2.

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16

Jepsen, Charles, and Roc Yang. "Making Squares from Pythagorean Triangles." College Mathematics Journal 29, no. 4 (1998): 284–88. http://dx.doi.org/10.1080/07468342.1998.11973956.

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17

Walser, Hans. "Lattice geometry and pythagorean triangles." Zentralblatt für Didaktik der Mathematik 32, no. 2 (2000): 32–35. http://dx.doi.org/10.1007/s11858-000-0001-8.

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18

Wen D. Chang and Russell A. Gordon. "Trisecting Angles in Pythagorean Triangles." American Mathematical Monthly 121, no. 7 (2014): 625. http://dx.doi.org/10.4169/amer.math.monthly.121.07.625.

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19

Baica, Malvina. "Pythagorean triangles of equal areas." International Journal of Mathematics and Mathematical Sciences 11, no. 4 (1988): 769–80. http://dx.doi.org/10.1155/s0161171288000948.

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The main intent in this paper is to find triples of Rational Pythagorean Triangles (abbr. RPT) having equal areas. A new method of solvinga2+ab+b2=c2is to seta=y−1,b=y+1,y∈N−{0,1}and get Pell's equationc2−3y2=1. To solvea2−ab−b2=c2, we seta=12(y+1),b=y−1,y≥2,y∈Nand get a corresponding Pell's equation. The infinite number of solutions in Pell's equation gives rise to an infinity of solutions toa2±ab+b2=c2. From this fact the following theorems are proved.Theorem 1 Letc2=a2+ab+b2,a+b>c>b>a>0, then the three RPT-s formed by(c,a),(c,b),(a+b,c)have the same areaS=abc(b+a)and there are i

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20

Euser, Martin. "Pythagorean Triangles and Musical Proportions." Nexus Network Journal 2, no. 1-2 (2000): 33–40. http://dx.doi.org/10.1007/s00004-999-0006-8.

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21

Wulandari, Hilaria Yesieka Ayu, and Tatag Yuli Eko Siswono. "Students Activities in Learning Pythagoras Theorems Using Desmos Application." Journal of Mathematical Pedagogy (JoMP) 5, no. 1 (2024): 1–14. http://dx.doi.org/10.26740/jomp.v5n1.p1-14.

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This study aims to describe the creative thinking skills of students in solving problems related to the Pythagorean Theorem using Desmos application. This descriptive research involved 3 Junior High School students in grade 9 in Sidoarjo (Indonesia) who each had high, medium, and low ability backgrounds. The task in the form of questions consisted of two questions asking students to create two different triangles. The results of the analysis showed that high-ability students were able to create two different triangles if they were given sides 3 and 4 in the form of a right triangle and an equi

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22

Thibodeau, Philip. "The Dependence of Ancient Greek Geometry and Metaphysics on Craft-Culture." Aestimatio: Sources and Studies in the History of Science 1 (April 30, 2021): 173–82. http://dx.doi.org/10.33137/aestimatio.v1i1.37624.

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A discussion of Robert Hahn’s The Metaphysics of the Pythagorean Theorem: Thales, Pythagoras, Engineering, Diagrams, and the Construction of the Cosmos out of Right Triangles. Published Online (2021-04-30)Copyright © 2021 by Philip Thibodeau Article PDF Link: https://jps.library.utoronto.ca/index.php/aestimatio/article/view/37624/28620 Corresponding Author: Philip Thibodeau,Brooklyn CollegeE-Mail: pthibodeau@brooklyn.cuny.edu

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23

Dr., R. Sivaraman. "TRIANGLE OF TRIANGULAR NUMBERS." TRIANGLE OF TRIANGULAR NUMBERS 9, no. 10 (2021): 2390–94. https://doi.org/10.47191/ijmcr/v9i10.01.

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Among several interesting number triangles that exist in mathematics, Pascal’s triangle is one of the best triangle possessing rich mathematical properties. In this paper, I will introduce a number triangle containing triangular numbers arranged in particular fashion. Using this number triangle, I had proved five interesting theorems which help us to generate Pythagorean triples as well as establish bijection between whole numbers and set of all integers.

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24

Regandla, Kriti, Ishra Zaman, and John Leddo. "An Investigation of Whether Exposure to Problems Involving Real-world Applications of the Pythagorean Theorem Facilitates Transfer of Learning." International Journal of Social Science and Economic Research 09, no. 12 (2024): 6010–16. https://doi.org/10.46609/ijsser.2024.v09i12.024.

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Transfer of learning is a fascinating concept that has intrigued educators and researchers for several years. Transfer of learning refers to the transferability of skills and knowledge from one scenario to another unfamiliar scenario. The present study investigates whether giving students real world examples of the application of the Pythagorean Theorem to real world settings would lead to improved performance on such problems on a post-test compared to giving students problems that only apply the Pythagorean Theorem to triangles. Twenty-one middle school students ranging from 6th to 8th grade

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25

K. Rajput, Chetansing. "A Classical Geometric Relationship That Reveals The Golden Link in Nature." JOURNAL OF ADVANCES IN MATHEMATICS 17 (December 11, 2019): 401–13. http://dx.doi.org/10.24297/jam.v17i0.8498.

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This paper introduces the perfect complementary relationship between the 3-4-5 Pythagorean triangle and the 1:2: right-angled triangle. The classical geometric intimacy between these two right triangles not only provides for the ultimate geometric substantiation of Golden Ratio, but it also reveals the fundamental Pi: Phi correlation (π: φ), with an extreme level of precision, and which is firmly based upon the classical geometric principles.

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26

Mustafa, Abdur Rahman, Dzuwi Nafida Hasan, Erna Zeniatul Baiti Albahria, et al. "Kabes Pupphy Learning Media (Pythagorean Proving Puzzle) for Mathematics Learning Pythagorean Material." Journal International Inspire Education Technology 1, no. 3 (2023): 8–16. http://dx.doi.org/10.55849/jiiet.v1i3.143.

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This study aims to describe learning media to facilitate students in constructing an understanding of the Pythagorean theorem. This study describes a learning media product in the form of a puzzle called Pupphy (Pythagorean Proving Puzzle). Pupphy is a puzzle proving the Pythagorean theorem on the Pythagorean theorem material for grade VIII students in junior high school. This media is different from the others because it is made from environmentally friendly materials that are easily found in the surrounding environment and can also be made by students themselves. This research uses a qualita

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27

Anglin, W. S. "Using Pythagorean Triangles to Approximate Angles." American Mathematical Monthly 95, no. 6 (1988): 540. http://dx.doi.org/10.2307/2322760.

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28

Akhtar, Mohammad Shakil. "86.47 Inscribed Circles of Pythagorean Triangles." Mathematical Gazette 86, no. 506 (2002): 302. http://dx.doi.org/10.2307/3621870.

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29

Navas, Andrés. "The Pythagorean Theorem Via Equilateral Triangles." Mathematics Magazine 93, no. 5 (2020): 343–46. http://dx.doi.org/10.1080/0025570x.2020.1817696.

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30

Anglin, W. S. "Using Pythagorean Triangles to Approximate Angles." American Mathematical Monthly 95, no. 6 (1988): 540–41. http://dx.doi.org/10.1080/00029890.1988.11972043.

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31

Nidhi, Sonam. "Exploring Properties and Classifications of Heron Triangles in Mathematics." Journal of Educational Research and Policies 7, no. 2 (2025): 89–99. https://doi.org/10.53469/jerp.2025.07(02).19.

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This study explores Heron triangles, which are unique triangles characterized by integer side lengths and areas. It classifies various types, including Pythagorean, consecutive, isosceles, and integral Heron triangles, and examines their mathematical properties such as semiperimeters, inradii, circumscribed radii, and heights. By presenting definitions, illustrations, and detailed proofs of theorems, this paper aims to deepen understanding and provide a foundation for further mathematical applications.

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32

Zhai, Wenguang. "On the number of primitive Pythagorean triangles." Acta Arithmetica 105, no. 4 (2002): 387–403. http://dx.doi.org/10.4064/aa105-4-6.

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33

Pambuccian, Victor, and Robin Chapman. "Pythagorean Triangles Are Not Quite Perfect: 11122." American Mathematical Monthly 113, no. 9 (2006): 848. http://dx.doi.org/10.2307/27642076.

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34

Jeyakrishnan, G., and G. Komahan. "Pentagonal Numbers, Heptagonal Numbers and Pythagorean Triangles." Journal of Mathematics and Informatics 11 (December 11, 2017): 17–19. http://dx.doi.org/10.22457/jmi.v11a3.

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35

Liu, Kui. "On the distribution of primitive Pythagorean triangles." Acta Arithmetica 144, no. 2 (2010): 135–50. http://dx.doi.org/10.4064/aa144-2-3.

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36

Bernstein, Leon. "On Primitive Pythagorean Triangles with Equal Perimeters." Fibonacci Quarterly 27, no. 1 (1989): 2–6. http://dx.doi.org/10.1080/00150517.1989.12429592.

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37

Benito, Manuel, and Juan L. Varona. "Pythagorean triangles with legs less than n." Journal of Computational and Applied Mathematics 143, no. 1 (2002): 117–26. http://dx.doi.org/10.1016/s0377-0427(01)00496-4.

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38

Maynard, Philip. "89.01 Perfect, and almost perfect, Pythagorean triangles." Mathematical Gazette 89, no. 514 (2005): 36–39. http://dx.doi.org/10.1017/s0025557200176636.

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39

Menzer, Hartmut. "On the Number of Primitive Pythagorean Triangles." Mathematische Nachrichten 128, no. 1 (1986): 129–33. http://dx.doi.org/10.1002/mana.19861280111.

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40

Amato, Roberto. "A characterization of Pythagorean triples." JP Journal of Algebra, Number Theory and Applications 39, no. 2 (2017): 221–30. https://doi.org/10.17654/NT039020221.

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The main aim of this paper is to present an analytic result which characterizes the Pythagorean triples via a cathetus. This way has the convenience to find easily all Pythagorean triples x, y, z ∈ ℕ, where x is a predetermined integer, which means finding all right triangles whose sides have integer measures and one cathetus is predetermined.

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41

Darbari, Mita. "A Connection between Hardy-Ramanujan Number and Special Pythagorean Triangles." Bulletin of Society for Mathematical Services and Standards 10 (June 2014): 45–47. http://dx.doi.org/10.18052/www.scipress.com/bsmass.10.45.

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42

Prabhakar, Giri. "Extending the plane trigonometric proof of Fermat’s Last Theorem to the case n = 3." Notes on Number Theory and Discrete Mathematics 31, no. 2 (2025): 410–28. https://doi.org/10.7546/nntdm.2025.31.2.410-428.

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We extend the plane trigonometric approach that we used to prove the case $n = 4$ of Fermat's Last Theorem, to the case $n=3.$ We show that all real positive triplets satisfying $a^\phi + b^\phi = c^\phi$ for $\phi > 1$ are triangles. As in the case of $n=4,$ we equate the Pythagorean and Fermat descriptions of the triangles for a particular smaller side while fixing the other sides, with $\phi=n$ being any positive integer. We hence show the existence of Ferma<span class="fontstyle0">t–</span>Pythagoras polynomials for $n \ge 3.$ For the case $n=3,$ we explicitly derive an anal

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43

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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44

Janaki, G. "Pythagorean Triangle with Area/Perimeter as a Disarium Number of Order 2 to 4." International Journal for Research in Applied Science and Engineering Technology 11, no. 5 (2023): 7573–75. http://dx.doi.org/10.22214/ijraset.2023.53485.

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45

Ericksen, Donna, John Stasiuk, and Martha Frank. "Sharing Teaching Ideas: Bringing Pythagoras to Life." Mathematics Teacher 88, no. 9 (1995): 744–47. http://dx.doi.org/10.5951/mt.88.9.0744.

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The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) states that “[o]ne of the most important properties in geometry, the Pythagorean theorem, is introduced in the middle grades” (p. 113). Although the Standards document assigns much prominence to the Pythagorean theorem, our experience teaching at the university level has revealed that students know the theorem by name and can recite a2 + b2 = c2 but that they often cannot handle even simple computations using the formula. Students' experience with the Pythagorean theorem in high school needs to be broadened by their con

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46

Bhanotar, Shailesh A., and Pradeep J. Jha. "Equivalent Statements and Conjectures Associated with Pythagorean Triangles." International Journal of Computational and Applied Mathematics 12, no. 3 (2017): 717. http://dx.doi.org/10.37622/ijcam/12.3.2017.717-727.

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47

Ramprasad, D. "Pythagorean triangles and addition of nonagonal, triangular numbers." Malaya Journal of Matematik 9, no. 1 (2021): 639–40. http://dx.doi.org/10.26637/mjm0901/0110.

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48

Heideman, Nic. "88.48 Escribed and inscribed circles of Pythagorean triangles." Mathematical Gazette 88, no. 512 (2004): 305–7. http://dx.doi.org/10.1017/s0025557200175163.

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49

Ehrman, Max. "Almost Primes in Thin Orbits of Pythagorean Triangles." International Mathematics Research Notices 2019, no. 11 (2017): 3498–526. http://dx.doi.org/10.1093/imrn/rnx191.

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Abstract Let $F=x^2+y^2-z^2$, and let $x_0 \in \mathbb{Z}^3$ be a $primitive$ solution to $F(x_0)=0$, e.g., so that its coordinates share no nontrivial divisor. Let $\Gamma \leq \mathrm{SO_F(\mathbb{Z})}$ be a thin subgroup. We consider the resulting thin orbits of Pythagorean triples $x_0 \cdot \Gamma$—specifically which hypotenuses, areas, and products of all three coordinates arise. We produce infinitely many $R$-almost primes in these three cases whenever $\Gamma$ has exponent $\delta_\Gamma&gt;\delta_0(R)$ for explicit $R$, $\delta_0$.

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50

SOYDAN, GOKHAN, YUSUF DOGRU, and N. UMUT ARSLANDOGAN. "The Pythagorean theorem and area formula for triangles on the plane with generalized absolute value metric." Creative Mathematics and Informatics 20, no. 1 (2011): 81–89. http://dx.doi.org/10.37193/cmi.2011.01.13.

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In this paper, we first give the Pythagorean theorem on the plane with generalized absolute value metric and show that the converse of the Pythagorean theorem is not true in this plane. Secondly, we give necessary and sufficient conditions for a triangle in this plane to have a right angle. Finally, we give a formula for the area of a triangle on this plane.

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