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Journal articles on the topic 'Pythagorean Triples'

Author: Grafiati

Published: 4 June 2021

Last updated: 12 February 2022

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1

Ochieng, Raymond Calvin, Chiteng’a John Chikunji, and Vitalis Onyango-Otieno. "Pythagorean Triples with Common Sides." Journal of Mathematics 2019 (April 1, 2019): 1–8. http://dx.doi.org/10.1155/2019/4286517.

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There exist a finite number of Pythagorean triples that have a common leg. In this paper we derive the formulas that generate pairs of primitive Pythagorean triples with common legs and also show the process of how to determine all the primitive and nonprimitive Pythagorean triples for a given leg of a Pythagorean triple.

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2

Murru, Nadir, Marco Abrate, Stefano Barbero, and Umberto Cerruti. "Groups and monoids of Pythagorean triples connected to conics." Open Mathematics 15, no. 1 (November 10, 2017): 1323–31. http://dx.doi.org/10.1515/math-2017-0111.

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Abstract We define operations that give the set of all Pythagorean triples a structure of commutative monoid. In particular, we define these operations by using injections between integer triples and 3 × 3 matrices. Firstly, we completely characterize these injections that yield commutative monoids of integer triples. Secondly, we determine commutative monoids of Pythagorean triples characterizing some Pythagorean triple preserving matrices. Moreover, this study offers unexpectedly an original connection with groups over conics. Using this connection, we determine groups composed by Pythagorean triples with the studied operations.

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3

Tikoo, Mohan, and Haohao Wang. "Generalized Pythagorean Triples and Pythagorean Triple Preserving Matrices." Missouri Journal of Mathematical Sciences 21, no. 1 (February 2009): 3–12. http://dx.doi.org/10.35834/mjms/1316032675.

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4

Agarwal, Ravi P. "Pythagorean Triples before and after Pythagoras." Computation 8, no. 3 (July 7, 2020): 62. http://dx.doi.org/10.3390/computation8030062.

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Following the corrected chronology of ancient Hindu scientists/mathematicians, in this article, a sincere effort is made to report the origin of Pythagorean triples. We shall account for the development of these triples from the period of their origin and list some known astonishing directions. Although for researchers in this field, there is not much that is new in this article, we genuinely hope students and teachers of mathematics will enjoy this article and search for new directions/patterns.

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5

Tirman, Alvin. "Pythagorean Triples." Mathematics Teacher 79, no. 8 (November 1986): 652–55. http://dx.doi.org/10.5951/mt.79.8.0652.

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If a class of students completing a course in plane geometry is asked to cite a few examples of primitive Pythagorean triples (those whose greatest common divisor is 1), their answers can be assumed to be 3, 4, 5; 5, 12, 13; 8. 15, 17: and 7, 24, 25.

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6

Koh, Wonjin, Geonwoo Kim, and Daeyeol Jeon. "Triangular Pythagorean Triples." Korean Science Education Society for the Gifted 9, no. 3 (December 26, 2017): 171–77. http://dx.doi.org/10.29306/jseg.2017.9.3.171.

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7

Evans, Chris. "75.23 Pythagorean Triples." Mathematical Gazette 75, no. 473 (October 1991): 317. http://dx.doi.org/10.2307/3619493.

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8

Eckert, Ernest J. "Primitive Pythagorean Triples." College Mathematics Journal 23, no. 5 (November 1992): 413. http://dx.doi.org/10.2307/2686417.

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9

Hildebrand, W. J. "Generalized Pythagorean Triples." College Mathematics Journal 16, no. 1 (January 1985): 48. http://dx.doi.org/10.2307/2686628.

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10

Frink, Orrin. "Almost Pythagorean Triples." Mathematics Magazine 60, no. 4 (October 1, 1987): 234. http://dx.doi.org/10.2307/2689346.

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11

Glasco, Shawn. "79.60 Pythagorean Triples." Mathematical Gazette 79, no. 486 (November 1995): 574. http://dx.doi.org/10.2307/3618100.

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12

Glaister, P. "Uncovering Pythagorean triples." Teaching Mathematics and its Applications: An International Journal of the IMA 25, no. 2 (June 1, 2006): 53–57. http://dx.doi.org/10.1093/teamat/hri003.

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13

Frink, Orrin. "Almost Pythagorean Triples." Mathematics Magazine 60, no. 4 (October 1987): 234–36. http://dx.doi.org/10.1080/0025570x.1987.11977310.

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14

Hildebrand, W. J. "Generalized Pythagorean Triples." College Mathematics Journal 16, no. 1 (January 1985): 48–52. http://dx.doi.org/10.1080/07468342.1985.11972849.

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15

Eckert, Ernest J. "Primitive Pythagorean Triples." College Mathematics Journal 23, no. 5 (November 1992): 413–17. http://dx.doi.org/10.1080/07468342.1992.11973493.

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16

Padavala, Pranay. "Pythagorean Triples Generator." Journal of Mathematics Research 13, no. 3 (May 26, 2021): 63. http://dx.doi.org/10.5539/jmr.v13n3p63.

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Given a leg of a right-angled triangle a, this formula gives the other leg b and the hypotenuse c by the usage of a pattern observed in Pythagorean triples. This is different as compared to Euclids method since Euclids method takes two arbitrary numbers as the input while this uses the side of the right-angled triangle as the input.

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17

Sporn, Howard. "A group of Pythagorean triples using the inradius." Mathematical Gazette 105, no. 563 (June 21, 2021): 209–15. http://dx.doi.org/10.1017/mag.2021.48.

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Pythagorean triples are triples of integers (a, b, c) satisfying the equation a2 + b2 = c2. For the purpose of this paper, we will take a, b and c to be positive, unless otherwise stated. Then, of course, it follows that a triple represents the lengths of sides of a right triangle. Also, for the purpose of this paper, we will consider the triples (a, b, c) and (b, a, c) to be distinct, even though they represent the same right triangle. A primitive Pythagorean triple is one for which a, b and c are relatively prime.

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18

Lefton, Phyllis. "A Matrix Method for Generating Pythagorean Triples." Mathematics Teacher 80, no. 2 (February 1987): 103–8. http://dx.doi.org/10.5951/mt.80.2.0103.

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This article describes a program that uses an interesting matrix method to generate Pythagorean triples -that is, solutions of the equation a2 + b2 = c2 for which a, b, and c are integers. Only primitive triples are found, that is, those for which a > 0, b > 0, c > 0, and the greatest common divisor of a, b, and c is one. This result suffices because nonprimitive triples are just multiples of primitive ones. We shall use the abbreviation PPT for primitive Pythagorean triple.

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19

Casinillo, Leomarich F., and Emily L. Casinillo. "SOME NOTES ON A GENERALIZED VERSION OF PYTHAGOREAN TRIPLES." Jurnal Riset dan Aplikasi Matematika (JRAM) 4, no. 2 (October 21, 2020): 103. http://dx.doi.org/10.26740/jram.v4n2.p103-107.

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A Pythagorean triple is a set of three positive integers a, b and c that satisfy the Diophantine equation a^2+b^2=c^2. The triple is said to be primitive if gcd(a, b, c)=1 and each pair of integers and are relatively prime, otherwise known as non-primitive. In this paper, the generalized version of the formula that generates primitive and non-primitive Pythagorean triples that depends on two positive integers k and n, that is, P_T=(a(k, n), b(k, n), c(k, n)) were constructed. Further, we determined the values of k and n that generates primitive Pythagorean triples and give some important results.

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20

Kalman, Dan. "Angling for Pythagorean Triples." College Mathematics Journal 17, no. 2 (March 1986): 167. http://dx.doi.org/10.2307/2686837.

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21

Spohn, William G., and Jeremy E. Dawson. "Neighborly Pythagorean Triples: 10704." American Mathematical Monthly 107, no. 9 (November 2000): 865. http://dx.doi.org/10.2307/2695752.

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22

Fassler, Albert. "Multiple Pythagorean Number Triples." American Mathematical Monthly 98, no. 6 (June 1991): 505. http://dx.doi.org/10.2307/2324870.

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23

Bhanu, K. S., and M. N. Deshpande. "96.02 Dual Pythagorean triples." Mathematical Gazette 96, no. 535 (March 2012): 96–99. http://dx.doi.org/10.1017/s0025557200004022.

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24

Jiang, Sheng, and Rui‐Chen Chen. "Digits reversed Pythagorean triples." International Journal of Mathematical Education in Science and Technology 29, no. 5 (September 1998): 689–96. http://dx.doi.org/10.1080/0020739980290505.

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25

Fässler, Albert. "Multiple Pythagorean Number Triples." American Mathematical Monthly 98, no. 6 (June 1991): 505–17. http://dx.doi.org/10.1080/00029890.1991.11995748.

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26

Kalman, Dan. "Angling for Pythagorean Triples." College Mathematics Journal 17, no. 2 (March 1986): 167–68. http://dx.doi.org/10.1080/07468342.1986.11972949.

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27

Arnold, Maxim, and Anatoly Eydelzon. "On Matrix Pythagorean Triples." American Mathematical Monthly 126, no. 2 (February 7, 2019): 158–60. http://dx.doi.org/10.1080/00029890.2019.1537760.

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28

Russell A. Gordon. "Semiperimeter and Pythagorean Triples." American Mathematical Monthly 118, no. 8 (2011): 680. http://dx.doi.org/10.4169/amer.math.monthly.118.08.680.

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29

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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30

DiDomenico, Angelo S., and Randy J. Tanner. "Pythagorean Triples from Harmonic Sequences." Mathematics Teacher 94, no. 3 (March 2001): 218–22. http://dx.doi.org/10.5951/mt.94.3.0218.

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Pythagorean triples have intrigued generations of mathematics explorers, including students, since ancient times. One of their most charming features is their connection with various other areas of mathematics. In the Mathematics Teacher, for example, authors have shown that Pythagorean triples can be generated from the Fibonacci numbers (Bertucci 1991), from geometric sequences (Carbeau 1993), and from both the addition and multiplication tables of whole numbers (DiDomenico 1993, 1995). These findings are indeed fascinating; when shared with students, they spark interest and curiosity and lead to a truly enriching mathematical experience. Students, in fact, independently found that Pythagorean triples could be generated from Fibonacci numbers and geometric sequences. This article reveals another surprising connection that shows how all primitive Pythagorean triples can be generated from harmonic sequences.

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31

Didomenico, Angelo S. "Pythagorean Triples from the Addition Table." Mathematics Teacher 78, no. 5 (May 1985): 346–48. http://dx.doi.org/10.5951/mt.78.5.0346.

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Open investigations are challenging and instructive and can generate considerable excitement. They can also lead to some unexpected and fascinating findings. In “Eureka! Pythagorean Triples from the Multiplication Table” (January 1983), I showed how one such investigation led to the derivation of all Pythagorean triples from simple properties of the multiplication table. Here we shall explore the addition table for positive integers and show that all Pythagorean triples can also be derived from simple patterns within this table.

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32

Sivaraman, R. "PASCAL TRIANGLE AND PYTHAGOREAN TRIPLES." International Journal of Engineering Technologies and Management Research 8, no. 8 (September 1, 2021): 75–80. http://dx.doi.org/10.29121/ijetmr.v8.i8.2021.1020.

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The concept of Pascal’s triangle has fascinated mathematicians for several centuries. Similarly, the idea of Pythagorean triples prevailing for more than two millennia continue to surprise even today with its abundant properties and generalizations. In this paper, I have demonstrated ways through four theorems to determine Pythagorean triples using entries from Pascal’s triangle.

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33

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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34

Stocks, Collin RM, and Gerald Lamb. "Delving Deeper: Calculating Pythagorean Triples." Mathematics Teacher 104, no. 2 (September 2010): 152–55. http://dx.doi.org/10.5951/mt.104.2.0152.

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35

Chang, Sungkon. "Direct parametrization of Pythagorean triples." Notes on Number Theory and Discrete Mathematics 25, no. 3 (September 30, 2019): 21–35. http://dx.doi.org/10.7546/nntdm.2019.25.3.21-35.

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36

McCullough, Darryl, and Elizabeth Wade. "Recursive Enumeration of Pythagorean Triples." College Mathematics Journal 34, no. 2 (March 2003): 107. http://dx.doi.org/10.2307/3595782.

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37

Dawson, Bryan. "A Ring of Pythagorean Triples." Missouri Journal of Mathematical Sciences 6, no. 2 (May 1994): 72–77. http://dx.doi.org/10.35834/1994/0602072.

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38

Chorlton, Frank. "81.29 Products of Pythagorean Triples." Mathematical Gazette 81, no. 491 (July 1997): 273. http://dx.doi.org/10.2307/3619209.

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39

Koshy, Thomas. "86.61 Generalised Fibonacci Pythagorean Triples." Mathematical Gazette 86, no. 507 (November 2002): 459. http://dx.doi.org/10.2307/3621140.

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40

Tirman, Alvin. "Geometric Parametrization of Pythagorean Triples." College Mathematics Journal 17, no. 2 (March 1986): 168. http://dx.doi.org/10.2307/2686838.

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41

Beauregard, Raymond A., and E. R. Suryanarayan. "Pythagorean Triples: The Hyperbolic View." College Mathematics Journal 27, no. 3 (May 1996): 170. http://dx.doi.org/10.2307/2687163.

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42

Wade, Peter W., and William R. Wade. "Recursions That Produce Pythagorean Triples." College Mathematics Journal 31, no. 2 (March 2000): 98. http://dx.doi.org/10.2307/2687578.

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43

Edwards, Thomas G. "Pythagorean Triples Served for Dessert." Mathematics Teaching in the Middle School 5, no. 7 (March 2000): 420–23. http://dx.doi.org/10.5951/mtms.5.7.0420.

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44

Gerstein, Larry J. "Pythagorean Triples and Inner Products." Mathematics Magazine 78, no. 3 (June 1, 2005): 205. http://dx.doi.org/10.2307/30044157.

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45

Scheffold, Egon. "Pythagorean Triples of Polygonal Numbers." American Mathematical Monthly 108, no. 3 (March 2001): 257. http://dx.doi.org/10.2307/2695388.

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46

MA, MI-MI, and YONG-GAO CHEN. "JEŚMANOWICZ’ CONJECTURE ON PYTHAGOREAN TRIPLES." Bulletin of the Australian Mathematical Society 96, no. 1 (March 13, 2017): 30–35. http://dx.doi.org/10.1017/s0004972717000107.

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In 1956, Jeśmanowicz conjectured that, for any positive integers $m$ and $n$ with $m>n$, $\gcd (m,n)=1$ and $2\nmid m+n$, the Diophantine equation $(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$ has only the positive integer solution $(x,y,z)=(2,2,2)$. In this paper, we prove the conjecture if $4\nmid mn$ and $y\geq 2$.

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47

Tong, Jingcheng. "87.56 Conjugates of Pythagorean triples." Mathematical Gazette 87, no. 510 (November 2003): 496–99. http://dx.doi.org/10.1017/s0025557200173711.

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48

Scheffold, Egon. "Pythagorean Triples of Polygonal Numbers." American Mathematical Monthly 108, no. 3 (March 2001): 257–58. http://dx.doi.org/10.1080/00029890.2001.11919749.

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49

Romik, Dan. "The dynamics of Pythagorean Triples." Transactions of the American Mathematical Society 360, no. 11 (April 22, 2008): 6045–64. http://dx.doi.org/10.1090/s0002-9947-08-04467-x.

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50

Tirman, Alvin. "Geometric Parametrization of Pythagorean Triples." College Mathematics Journal 17, no. 2 (March 1986): 168. http://dx.doi.org/10.1080/07468342.1986.11972950.

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