Applied and Computational Mathematics

| Peer-Reviewed |

Theorem on a Matrix of Right-Angled Triangles

Received: 11 March 2013    Accepted:     Published: 02 April 2013
Views:       Downloads:

Share This Article

Abstract

The following theorem is proved: All primitive right-angled triangles (primitive Pythagorean triples) may be defined by a pair of positive integer indices (i,j), where i is an uneven number and j is an even number and have no com-mon factor. The sides of every positive integer right angled triangle are then defined by the indices as follows: For hy-potenuse h, uneven leg u and even leg e, h = i2 + ij + j2/2, e = ij + j2/2, u = i2 + ij. This defines an infinite by infinite matrix of right angled triangles with positive integer sides.

DOI 10.11648/j.acm.20130202.14
Published in Applied and Computational Mathematics (Volume 2, Issue 2, April 2013)
Page(s) 36-41
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Primitive Right-Angled Triangles, Pythagorean Triples, Infinite Two-Dimensional Matrix

References
[1] ES Rowland, "Pythagorean Triples Project," http://www. google.co.th/search?sourceid=navclient&aq=hts&oq=&ie=UTF-8&rlz=1T4ADRA_enTH433TH434&q=Pythagorean+ Triples+Project
[2] Wikipedia, "Generating Pythagorean Triples," http://en. wikipedia.org/wiki/Formulas_for_generating_Pythagorean_ triples
[3] R Simms, "Pythagorean Triples," http://www.math.clemson. edu/~simms/neat/math/pyth/
[4] LP Fibonacci, Liber Quadratorum, 1225.
[5] LP Fibonacci, The Book of Squares (Liber Quadratorum),. An annotated translation into modern English by LE Sigler, Academic Press, Orlando, FL, 1987 (ISBN 978-0-12-643130-8)
[6] M Stifel, Arithmetica Integra, 1544.
[7] J. Ozanam, Recreations in Mathematics and Natural Phi-losophy, 1814.
[8] J. Ozanam, Science and Natural Philosophy: Dr. Hutton’s Translation of Montucla’s edition of Ozanam, 1844, revised by Edward Riddle, Thomas Tegg, London.
[9] Euclid's Elements: Book X, Proposition XXIX.
[10] LE Dickson, History of the Theory of Numbers, Vol II, Dio-phantine Analysis, 1920 (Carnegie Institution of Washington, Publication No 256).
Author Information
  • Department of Science, Asia-Pacific International University; PO Box 4, MuakLek, Saraburi Province, 18180, Thailand

Cite This Article
  • APA Style

    Martin W. Bredenkamp. (2013). Theorem on a Matrix of Right-Angled Triangles. Applied and Computational Mathematics, 2(2), 36-41. https://doi.org/10.11648/j.acm.20130202.14

    Copy | Download

    ACS Style

    Martin W. Bredenkamp. Theorem on a Matrix of Right-Angled Triangles. Appl. Comput. Math. 2013, 2(2), 36-41. doi: 10.11648/j.acm.20130202.14

    Copy | Download

    AMA Style

    Martin W. Bredenkamp. Theorem on a Matrix of Right-Angled Triangles. Appl Comput Math. 2013;2(2):36-41. doi: 10.11648/j.acm.20130202.14

    Copy | Download

  • @article{10.11648/j.acm.20130202.14,
      author = {Martin W. Bredenkamp},
      title = {Theorem on a Matrix of Right-Angled Triangles},
      journal = {Applied and Computational Mathematics},
      volume = {2},
      number = {2},
      pages = {36-41},
      doi = {10.11648/j.acm.20130202.14},
      url = {https://doi.org/10.11648/j.acm.20130202.14},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.acm.20130202.14},
      abstract = {The following theorem is proved:  All primitive right-angled triangles (primitive Pythagorean triples) may be defined by a pair of positive integer indices (i,j), where i is an uneven number and j is an even number and have no com-mon factor.  The sides of every positive integer right angled triangle are then defined by the indices as follows:  For hy-potenuse h, uneven leg u and even leg e, h = i2 + ij + j2/2, e = ij + j2/2, u = i2 + ij.  This defines an infinite by infinite matrix of right angled triangles with positive integer sides.},
     year = {2013}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Theorem on a Matrix of Right-Angled Triangles
    AU  - Martin W. Bredenkamp
    Y1  - 2013/04/02
    PY  - 2013
    N1  - https://doi.org/10.11648/j.acm.20130202.14
    DO  - 10.11648/j.acm.20130202.14
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
    SP  - 36
    EP  - 41
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20130202.14
    AB  - The following theorem is proved:  All primitive right-angled triangles (primitive Pythagorean triples) may be defined by a pair of positive integer indices (i,j), where i is an uneven number and j is an even number and have no com-mon factor.  The sides of every positive integer right angled triangle are then defined by the indices as follows:  For hy-potenuse h, uneven leg u and even leg e, h = i2 + ij + j2/2, e = ij + j2/2, u = i2 + ij.  This defines an infinite by infinite matrix of right angled triangles with positive integer sides.
    VL  - 2
    IS  - 2
    ER  - 

    Copy | Download

  • Sections