Why are some right-end digits absent in primitive Pythagorean triples?

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
Full paper (PDF, 171 Kb)

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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

  1. Dickson, L.E. 1952. History of the Theory of Numbers. Volume 1. New York: Chelsea.
  2. Hardy, G.H., E.M. Wright. 1965. An Introduction to the Theory of Numbers. Oxford: Clarendon Press.
  3. 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.
  4. Leyendekkers, J.V., A.G. Shannon, J.M. Rybak. 2007. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No 9.
  5. 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.
  6. 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.
  7. 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.

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