Theorem on a Matrix of Right-Angled Triangles

Martin W. Bredenkamp

doi:doi:10.11648/j.acm.20130202.14

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Theorem on a Matrix of Right-Angled Triangles

Martin W. Bredenkamp

Published in Applied and Computational Mathematics (Volume 2, Issue 2)

Published: 2 April 2013

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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.

Published in	Applied and Computational Mathematics (Volume 2, Issue 2)
DOI	10.11648/j.acm.20130202.14
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), 2013. 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).

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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

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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

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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

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@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://article.sciencepublishinggroup.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}
}

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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  -

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Author Information

Martin W. Bredenkamp

Department of Science, Asia-Pacific International University

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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://article.sciencepublishinggroup.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  -

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