Odd-powered triples and Pellian sequences

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 18, 2012, Number 3, Pages 8—12
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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

Even and odd integers of the form Nm in the modular ring Z4 have rows with elements which satisfy Pellian recurrence relations from which Pythagorean triples can form when m = 2. When m is odd, they have different and incompatible Pellian row structure and triples are not formed.

Keywords

  • Modular rings
  • Integer structure analysis
  • Pellian sequences
  • Pythagorean triples
  • Triangular numbers
  • Pentagonal numbers

AMS Classification

  • 11A41
  • 11A07

References

  1. Hardy, G. H. A Mathematician’s Apology. (Foreword by C.P. Snow.) Cambridge: University Press, 1967.
  2. Hardy, G. H., E. M. Wright. An Introduction to the Theory of Numbers, 3rd edition. London: Oxford University Press, 1954.
  3. Leyendekkers, J. V., A. G. Shannon, J. M. Rybak. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No. 9, 2007.
  4. Leyendekkers, J. V., A. G. Shannon. Why 3 and 5 are always Factors of Primitive Pythagorean Triples. International Journal of Mathematical Education in Science & Technology. Vol. 42, 2010, 102–105.
  5. Leyendekkers, J. V., A. G. Shannon. Rows of Odd Powers in the Modular Ring Z4. Notes on Number Theory and Discrete Mathematics. Vol. 16, 2010, No. 2, 24–32.
  6. Leyendekkers, J. V., A. G. Shannon. Modular Rings and the Integer 3. Notes on Number Theory and Discrete Mathematics. Vol. 17, 2011, No. 2, 47–51.
  7. Leyendekkers, J. V., A. G. Shannon. The Structure of Even Powers in Z3: Critical Structural Factors that Prevent the Formation of Even-powered Triples greater than Squares. Notes on Number Theory and Discrete Mathematics. Vol. 17, 2011, No. 3, 26–30.
  8. Leyendekkers, J. V., A. G. Shannon. The Structure of π. Notes on Number Theory and Discrete Mathematics. Vol. 17, 2011, No. 4, 61–68.
  9. Leyendekkers, J. V., A. G. Shannon. The Modular Ring Z5. Notes on Number Theory and Discrete Mathematics, Vol. 18, 2012, No. 2, 28–33.

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Cite this paper

APA

Leyendekkers, J. V., & Shannon, A. G. (2012). Odd-powered triples and Pellian sequences, Notes on Number Theory and Discrete Mathematics, 18(3), 8-12.

Chicago

Leyendekkers, J. V., and A. G. Shannon. “Odd-powered triples and Pellian sequences.” Notes on Number Theory and Discrete Mathematics 18, no. 3 (2012): 8-12.

MLA

Leyendekkers, J. V., and A. G. Shannon. “Odd-powered triples and Pellian sequences.” Notes on Number Theory and Discrete Mathematics 18.3 (2012): 8-12. Print.

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