Theorem on a Matrix of Right-Angled Triangles
Applied and Computational Mathematics
Volume 2, Issue 2, April 2013, Pages: 36-41
Received: Mar. 11, 2013; Published: Apr. 2, 2013
Views 2548      Downloads 131
Author
Martin W. Bredenkamp, Department of Science, Asia-Pacific International University; PO Box 4, MuakLek, Saraburi Province, 18180, Thailand
Article Tools
PDF
Follow on us
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.
Keywords
Primitive Right-Angled Triangles, Pythagorean Triples, Infinite Two-Dimensional Matrix
To cite this article
Martin W. Bredenkamp, Theorem on a Matrix of Right-Angled Triangles, Applied and Computational Mathematics. Vol. 2, No. 2, 2013, pp. 36-41. doi: 10.11648/j.acm.20130202.14
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).
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186