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Journal articles on the topic 'Primitive pythagorean triples'

Author: Grafiati

Published: 5 June 2025

Last updated: 2 August 2025

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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

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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3

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

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4

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

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5

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 (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 resul

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6

Lefton, Phyllis. "A Matrix Method for Generating Pythagorean Triples." Mathematics Teacher 80, no. 2 (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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7

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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8

Amato, Roberto. "A characterization of primitive Pythagorean triples." Palestine Journal of Mathematics 12, no. 2 (2023): 524–29. https://doi.org/10.5281/zenodo.10460794.

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The main aim of this paper is to present an analytic result which characterizes the primitive Pythagorean triples via a cathetus. This way has the convenience to find easily all primitive Pythagorean triples x, y, z ∈ N where x is a predetermined integer.

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9

Tirman, Alvin. "Pythagorean Triples." Mathematics Teacher 79, no. 8 (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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10

Austin, Jathan. "A note on generating primitive Pythagorean triples using matrices." Notes on Number Theory and Discrete Mathematics 29, no. 2 (2023): 402–6. http://dx.doi.org/10.7546/nntdm.2023.29.2.402-406.

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We present matrices that generate families of primitive Pythagorean triples that arise from generalized Fibonacci sequences. We then present similar results for generalized Lucas sequences and primitive Pythagorean triples.

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11

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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12

Sporn, Howard. "Pythagorean triples using the relativistic velocity addition formula." Mathematical Gazette 108, no. 572 (2024): 219–24. http://dx.doi.org/10.1017/mag.2024.60.

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In the special theory of relativity, there is an unusual formula for addition of velocities. We will use this formula to generate Pythagorean triples. We define a Pythagorean triple in the usual manner as a triple of positive integers (a, b, c) such that a2 + b2 = c2. The numbers a and b are called legs, and c is called the hypotenuse. Later, we will allow b to take on the value of any integer. A primitive Pythagorean triple is a Pythagorean triple for which a, b and c are relatively prime.

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Rajput, Chetansing, and Hariprasad Manjunath. "Metallic means and Pythagorean triples." Notes on Number Theory and Discrete Mathematics 30, no. 1 (2024): 184–94. http://dx.doi.org/10.7546/nntdm.2024.30.1.184-194.

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In this article, we study the connection between Pythagorean triples and metallic means. We derive several interconnecting identities between different metallic means. We study the Pythagorean triples in the three-term recurrent sequences corresponding to different metallic means. Further, we relate different families of primitive Pythagorean triples to the corresponding metallic means.

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14

Sporn, Howard. "A group of Pythagorean triples using the inradius." Mathematical Gazette 105, no. 563 (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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15

Ronald, Skurnick. "CONSTRUCTING INFINITE FAMILIES OF GENERALIZED PYTHAGOREAN TRIPLES." International Journal of Novel Research in Physics Chemistry & Mathematics 12, no. 1 (2025): 18–21. https://doi.org/10.5281/zenodo.14849015.

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<strong>Abstract:</strong> In this article, we show how to construct an infinite family of generalized Pythagorean triples.  Then, we consider how Fermat's Last Theorem relates to this construction. <strong>Keywords:</strong> Pythagorean triple, primitive Pythagorean triple, convergent infinite geometric series, Fermat's Last Theorem. <strong>Title:</strong> CONSTRUCTING INFINITE FAMILIES OF GENERALIZED PYTHAGOREAN TRIPLES <strong>Author:</strong> Ronald Skurnick <strong>International Journal of Novel Research in Physics Chemistry & Mathematics</strong> <strong>ISSN 2394-9651</strong>

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16

Kak, Subhash, and Monisha Prabhu. "Cryptographic Applications of Primitive Pythagorean Triples." Cryptologia 38, no. 3 (2014): 215–22. http://dx.doi.org/10.1080/01611194.2014.915257.

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17

Cross, James T. "Primitive Pythagorean Triples of Gaussian Integers." Mathematics Magazine 59, no. 2 (1986): 106–10. http://dx.doi.org/10.1080/0025570x.1986.11977231.

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18

Gil, Byung Keon, Ji-Woo Han, Tae Hyun Kim, et al. "Frobenius numbers of Pythagorean triples." International Journal of Number Theory 11, no. 02 (2015): 613–19. http://dx.doi.org/10.1142/s1793042115500323.

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Given relatively prime integers a1,…, an, the Frobenius number g(a1,…, an) is defined as the largest integer which cannot be expressed as x1a1 +⋯+ xnan with xi nonnegative integers. In this paper, we give the Frobenius number of primitive Pythagorean triples:

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19

DiDomenico, Angelo S., and Randy J. Tanner. "Pythagorean Triples from Harmonic Sequences." Mathematics Teacher 94, no. 3 (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 lea

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20

Antalan, John Rafael M., and Richard P. Tagle. "NUMERIC PALINDROMES IN PRIMITIVE AND NON PRIMITIVE PYTHAGOREAN TRIPLES." JP Journal of Algebra, Number Theory and Applications 37, no. 1 (2015): 21–30. http://dx.doi.org/10.17654/jpantaaug2015_021_030.

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21

Amato, Roberto. "GROUPS AND MONOID IN THE SET OF PYTHAGOREAN TRIPLES." INTEGERS 24, A5 (2024): 13. https://doi.org/10.5281/zenodo.10462697.

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The primary objective of this paper is to find suitable binary operations on the set of Pythagorean triples, obtaining two commutative infinite groups, one with elements in Q and the other with elements in Z. Additionally, we aim to get a commutativeinfinite monoid with elements in N or in Z. In particular, on the set of primitive Pythagorean triples, we establish a commutative infinite group with elements in Z.

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22

Terai, Nobuhiro. "On Jeśmanowicz' conjecture concerning primitive Pythagorean triples." Journal of Number Theory 141 (August 2014): 316–23. http://dx.doi.org/10.1016/j.jnt.2014.02.009.

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23

Bonsangue, Martin Vern. "A Geometrical Representation of Primitive Pythagorean Triples." Mathematics Teacher 90, no. 5 (1997): 350–54. http://dx.doi.org/10.5951/mt.90.5.0350.

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The geoboard has long been a useful tool for mathematics students of all ages to explore geometric patterns and relationships. Among the many interesting activities that can be done on geoboards is exploring properties of polygons, such as the relationship between a polygon's number of sides and the sum of its interior angles. Such software programs as The Geometer's Sketchpad (Jackiw 1990) can also be helpful in exploring these and even more sophisticated geometric relationships. These tools can be helpful not only in discovering known relationships but also in making new conjectures on the b

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24

Krylov, Nikolai, and Lindsay Kulzer. "The group of primitive almost pythagorean triples." Involve, a Journal of Mathematics 6, no. 1 (2013): 13–24. http://dx.doi.org/10.2140/involve.2013.6.13.

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25

Price, H. Lee, and Frank R. Bernhart. "Pythagoras' Garden, Revisited." Australian Senior Mathematics Journal 26, no. 1 (2012): 29–40. https://doi.org/10.5281/zenodo.3825040.

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Mack and Czernezkyj (2010) have given an interesting account of primitive Pythagorean triples (PPTs) from a geometrical perspective. In this article, the authors wish to enlarge on the role of the equicircles (incircle and three excircles), and show there is yet another family tree in Pythagoras' garden. Where Mack and Czernezkyj (2010) begin with four equicircles, the authors begin with four tangent circles, attached to the corners of a rectangle based on the right triangle. Reflecting these circles in a certain line results in a congruent tangent cluster, having the same six points of ta

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26

Terr, David. "Some Interesting Infinite Families of Primitive Pythagorean Triples." Fibonacci Quarterly 50, no. 1 (2012): 68–81. http://dx.doi.org/10.1080/00150517.2012.12428024.

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27

Miyazaki, T., and N. Terai. "On Jeśmanowicz’ conjecture concerning primitive Pythagorean triples. II." Acta Mathematica Hungarica 147, no. 2 (2015): 286–93. http://dx.doi.org/10.1007/s10474-015-0552-3.

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28

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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29

R. Sivaraman, R. Jayamari,. "Construction of Perfect Squares and Pythagorean Triples using Continued Fractions." Advances in Nonlinear Variational Inequalities 27, no. 1 (2024): 286–307. http://dx.doi.org/10.52783/anvi.v27.437.

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Continued fractions play a major role in Number theory and solving Diophantine equations. With the help of Continued fractions and Pell’s equations, we have proved some interesting results like obtaining Pythagorean triples whose smaller sides differ by some particular integer and also we have provided methods to generate primitive Pythagorean triples satisfying a specific condition. Perfect squares are one of the most well-known class of numbers in number theory. In this paper we introduced ways to produce natural numbers that are related to perfect squares with some specified conditions.

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30

MIYAZAKI, TAKAFUMI. "THE SHUFFLE VARIANT OF JEŚMANOWICZ’ CONJECTURE CONCERNING PYTHAGOREAN TRIPLES." Journal of the Australian Mathematical Society 90, no. 3 (2011): 355–70. http://dx.doi.org/10.1017/s1446788711001340.

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AbstractLet (a,b,c) be a primitive Pythagorean triple such that b is even. In 1956, Jeśmanowicz conjectured that the equation ax+by=cz has the unique solution (x,y,z)=(2,2,2) in the positive integers. This is one of the most famous unsolved problems on Pythagorean triples. In this paper we propose a similar problem (which we call the shuffle variant of Jeśmanowicz’ problem). Our problem states that the equation cx+by=az with x,y and z positive integers has the unique solution (x,y,z)=(1,1,2) if c=b+1 and has no solutions if c>b+1 . We prove that the shuffle variant of the Jeśmanowicz proble

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31

Wang, Jingzhe. "On Pythagorean Triples and the Primitive Roots Modulo a Prime." Journal of Mathematics 2021 (July 29, 2021): 1–8. http://dx.doi.org/10.1155/2021/7634728.

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In this paper, we use the elementary methods and the estimates for character sums to study a problem related to primitive roots and the Pythagorean triples and prove the following result: let p be an odd prime large enough. Then, there must exist three primitive roots x , y , and z modulo p such that x 2 + y 2 = z 2 .

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32

Bradley, Christopher J. "From integer Lorentz transformations to Pythagoras." Mathematical Gazette 88, no. 511 (2004): 16–21. http://dx.doi.org/10.1017/s0025557200174182.

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In this article we show how to obtain by recurrence all primitive Pythagorean triples from the single basic (4, 3, 5) triple. It is all done by repeated application of two integer unimodular transformations that leave the indefinite metric x2 + y2 - z2 invariant, together with the additional transformations (i) that change the sign of x and (ii) that change the sign of y. These alone would restrict the triples to those in which x is even, y is odd and z is positive, so we then include two further transformations (iii) that exchange x and y and (iv) that change the sign of z, thereby accounting

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33

Le, Maohua. "A note on Jeśmanowicz' conjecture concerning Pythagorean triples." Bulletin of the Australian Mathematical Society 59, no. 3 (1999): 477–80. http://dx.doi.org/10.1017/s0004972700033177.

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Let n be a positive integer, and let (a, b, c) be a primitive Pythagorean triple. In this paper we give certain conditions for the equation (an)x + (bn)y = (cn)z to have positive integer solutions (x, y, z) with (x, y, z) ≠ (2, 2, 2). In particular, we show that x, y and z must be distinct.

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34

Beauregard, Raymond A., and E. R. Suryanarayan. "Proof without Words: Parametric Representation of Primitive Pythagorean Triples." Mathematics Magazine 69, no. 3 (1996): 189. http://dx.doi.org/10.2307/2691466.

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35

Mitchell, Douglas W. "85.27 An Alternative Characterisation of All Primitive Pythagorean Triples." Mathematical Gazette 85, no. 503 (2001): 273. http://dx.doi.org/10.2307/3622017.

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Beauregard, Raymond A., and E. R. Suryanarayan. "Proof Without Words: Parametric Representation of Primitive Pythagorean Triples." Mathematics Magazine 69, no. 3 (1996): 189. http://dx.doi.org/10.1080/0025570x.1996.11996425.

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37

DENG, MOU-JIE, and DONG-MING HUANG. "A NOTE ON JEŚMANOWICZ’ CONJECTURE CONCERNING PRIMITIVE PYTHAGOREAN TRIPLES." Bulletin of the Australian Mathematical Society 95, no. 1 (2016): 5–13. http://dx.doi.org/10.1017/s0004972716000605.

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Let $a,b,c$ be a primitive Pythagorean triple and set $a=m^{2}-n^{2},b=2mn,c=m^{2}+n^{2}$, where $m$ and $n$ are positive integers with $m>n$, $\text{gcd}(m,n)=1$ and $m\not \equiv n~(\text{mod}~2)$. In 1956, Jeśmanowicz conjectured that the only positive integer solution to the Diophantine equation $(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$ is $(x,y,z)=(2,2,2)$. We use biquadratic character theory to investigate the case with $(m,n)\equiv (2,3)~(\text{mod}~4)$. We show that Jeśmanowicz’ conjecture is true in this case if $m+n\not \equiv 1~(\text{mod}~16)$ or $y>1$. Finally, using t

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38

Yang, Hai, and Ruiqin Fu. "A note on Jeśmanowicz' conjecture concerning primitive Pythagorean triples." Journal of Number Theory 156 (November 2015): 183–94. http://dx.doi.org/10.1016/j.jnt.2015.04.009.

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39

Chen, Xiao-Yan, and Mou-Jie Deng. "A NOTE ON JEŚMANOWICZ’ CONJECTURE CONCERNING PRIMITIVE PYTHAGOREAN TRIPLES." JP Journal of Algebra, Number Theory and Applications 37, no. 1 (2015): 31–40. http://dx.doi.org/10.17654/jpantaaug2015_031_040.

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40

Nguyen, Van Thien, Viet Kh Nguyen, and Pham Hung Quy. "A note on Jeśmanowicz’ conjecture for non-primitive Pythagorean triples." Open Journal of Mathematical Sciences 5, no. 1 (2021): 115–27. http://dx.doi.org/10.30538/oms2021.0150.

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Let \((a, b, c)\) be a primitive Pythagorean triple parameterized as \(a=u^2-v^2, b=2uv, c=u^2+v^2\), where \(u>v>0\) are co-prime and not of the same parity. In 1956, L. Jesmanowicz conjectured that for any positive integer \(n\), the Diophantine equation \((an)^x+(bn)^y=(cn)^z\) has only the positive integer solution \((x,y,z)=(2,2,2)\). In this connection we call a positive integer solution \((x,y,z)\ne (2,2,2)\) with \(n>1\) exceptional. In 1999 M.-H. Le gave necessary conditions for the existence of exceptional solutions which were refined recently by H. Yang and R.-Q. Fu. In thi

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41

Mani, K., and A. Devi. "Modified DES using Different Keystreams Based On Primitive Pythagorean Triples." International Journal of Mathematical Sciences and Computing 3, no. 1 (2017): 38–48. http://dx.doi.org/10.5815/ijmsc.2017.01.04.

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42

Yang, Hai, and Ruiqin Fu. "An open problem on Jeśmanowicz' conjecture concerning primitive Pythagorean triples." Glasnik Matematicki 54, no. 2 (2019): 271–77. http://dx.doi.org/10.3336/gm.54.2.02.

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43

Deng, M. J., and J. Guo. "A note on Jeśmanowicz’ conjecture concerning primitive Pythagorean triples. II." Acta Mathematica Hungarica 153, no. 2 (2017): 436–48. http://dx.doi.org/10.1007/s10474-017-0751-1.

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44

Leyendekkers, J. V., and A. G. Shannon. "Why 3 and 5 are always factors of primitive Pythagorean triples." International Journal of Mathematical Education in Science and Technology 42, no. 1 (2011): 102–5. http://dx.doi.org/10.1080/0020739x.2010.510219.

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45

Okagbue, Hilary I., Muminu O. Adamu, Pelumi E. Oguntunde, Abiodun A. Opanuga, Enahoro A. Owoloko, and Sheila A. Bishop. "Datasets on the statistical and algebraic properties of primitive Pythagorean triples." Data in Brief 14 (October 2017): 686–94. http://dx.doi.org/10.1016/j.dib.2017.08.021.

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46

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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47

Le, Maohua, and Gökhan Soydan. "An application of Baker’s method to the Jeśmanowicz’ conjecture on primitive Pythagorean triples." Periodica Mathematica Hungarica 80, no. 1 (2019): 74–80. http://dx.doi.org/10.1007/s10998-019-00295-0.

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48

Fu, Ruiqin, and Hai Yang. "A note on the exceptional solutions of Jeśmanowicz’ conjecture concerning primitive Pythagorean triples." Periodica Mathematica Hungarica 81, no. 2 (2020): 275–83. http://dx.doi.org/10.1007/s10998-020-00317-2.

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49

Firstov, V. E. "A special matrix transformation semigroup of primitive pairs and the genealogy of Pythagorean triples." Mathematical Notes 84, no. 1-2 (2008): 263–79. http://dx.doi.org/10.1134/s0001434608070262.

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Piyadasa, RAD, J. Munasinghe, DK Mallawa Arachchi, and KH Kumara. "A new interpretation of primitive Pythagorean triples and a conjecture related to Fermat's Last Theorem." Journal of Science of the University of Kelaniya Sri Lanka 3 (January 24, 2011): 93. http://dx.doi.org/10.4038/josuk.v3i0.2742.

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