J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 17, 2011, Number 1, Pages 37—44
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Authors and affiliations
J. V. Leyendekkers
Faculty of Science, The University of Sydney
Sydney, NSW 2006, Australia
A. G. Shannon
Faculty of Engineering & IT, University of Technology
Sydney, NSW 2007, Australia
Abstract
Integer structure analysis in the Ring Z3 shows that the right-end digit (RED) couples (1,4), (5,6) and (5,0) for x2, y2 in the primitive Pythagorean triple (pPt) in the equation z2 = x2 + y2 do not lead to the primitive form of triple. The rows of x2, y2 with these REDs do not add to the required form for z2. Since 3 ∤ z , the row of z2 must follow the pentagonal numbers. Common factors for x, y are also inconsistent with pPt formation so that the (x2, y2). RED (5,0) may be discarded directly.
Keywords
- Integer structure analysis
- Modular rings
- Right-end-digits
- Primitive Pythagorean triples
- Triangular numbers
- Pentagonal numbers
AMS Classification
- 11A41
- 11A07
References
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- Hardy, G.H., E.M. Wright. 1965. An Introduction to the Theory of Numbers. Oxford: Clarendon Press.
- Horadam, A.F., A.G. Shannon. 1988. Asveld’s Polynomials pj(n). In A.N. Philippou, A.F. Horadam, G.E. Bergum (eds). Applications of Fibonacci Numbers. Dordrecht: Kluwer, pp.163-176.
- Leyendekkers, J.V., A.G. Shannon, J.M. Rybak. 2007. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No 9.
- Leyendekkers, J.V., A.G. Shannon. 2011. The Number of Primitive Pythagorean Triples in a Given Interval. Notes on Number Theory & Discrete Mathematics In press.
- Leyendekkers, J.V., A.G. Shannon. 2011. Why 3 and 5 are always Factors of Primitive Pythagorean Triples. International Journal of Mathematical Education in Science & Technology. 42 (1): 102-105.
- Leyendekkers, J.V., A.G. Shannon. 2011. Modular Rings and the Integer 3. In preparation.
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Cite this paper
Leyendekkers, J., & Shannon, A.(2011). Why are some right-end digits absent in primitive Pythagorean triples?, Notes on Number Theory and Discrete Mathematics, 17(1), 37-44.