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
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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

APA

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.

Chicago

Leyendekkers, JV, and AG Shannon. “Why Are Some Right-end Digits Absent in Primitive Pythagorean Triples?” Notes on Number Theory and Discrete Mathematics 17, no. 1 (2011): 37-44.

MLA

Leyendekkers, JV, and AG Shannon. “Why Are Some Right-end Digits Absent in Primitive Pythagorean Triples?” Notes on Number Theory and Discrete Mathematics 17.1 (2011): 37-44. Print.

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