The structure of geometric number sequences

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 17, 2011, Number 3, Pages 31—37
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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

An integer structure analysis (ISA) of the triangular, tetrahedral, pentagonal and pyramidal numbers is developed. The relationships among the elements and the powers of the elements of these sequences are discussed. In particular, the triangular and pentagonal numbers are directly linked and are structurally important for the formation of triples. The class structure in the modular rings Z3 and Z4 of some elements of these sequences reinforce previous studies of their properties.

Keywords

  • Integer structure analysis
  • Modular rings
  • Prime numbers
  • Triangular numbers
  • Pentagonal numbers.
  • Octagonal numbers
  • Repunits

AMS Classification

  • 11A41
  • 11A07

References

  1. Atanassov, K., V. Atanassova, A. Shannon, J. Turner. New Visual Perspectives on Fibonacci Numbers. New Jersey, World Scientific, 2002.
  2. Ball, W. W. R., H. S. M. Coxeter. Mathematical Recreations and Essays. New York, Macmillan, 1956.
  3. Leyendekkers, J. V., A. G. Shannon, J. M. Rybak. Pattern Recognition: Modular Rings
    and Integer Structure. North Sydney: Raffles KvB Monograph, 2007, No. 9.
  4. Leyendekkers, J. V., A. G. Shannon. Integer Structure Analysis of the Product of Adjacent Integers and Euler’s Extension of Fermat’s Last Theorem. Advanced Studies in Contemporary Mathematics. Vol. 17, 2008, No. 2, 221–229.
  5. Leyendekkers, J. V., A. G. Shannon. Spectra of Primes. Proceedings of the Jangjeon Mathematical Society. Vol. 12, 2009, No. 1, 1–10.
  6. Leyendekkers, J. V., A. G. Shannon. Integer Structure Analysis of Odd Powered Triples: The Significance of Triangular versus Pentagonal Numbers. Notes on Number Theory and Discrete Mathematics. Vol. 16, 2010, No. 4, 6–13.
  7. 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, 29–32.
  8. Leyendekkers, J. V., A. G. Shannon. The Number of Primitive Pythagorean Triples in a Given Interval. Notes on Number Theory and Discrete Mathematics. In press.
  9. 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.
  10. Sloane, N. J. A., S. Plouffe. The Encyclopedia of Integer Sequences. San Diego, Academic Press, 1995.

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Cite this paper

APA

Leyendekkers, J., & Shannon, A.(2011). The structure of geometric number sequences, Notes on Number Theory and Discrete Mathematics, 17(3), 31-37.

Chicago

Leyendekkers, JV, and AG Shannon. “The Structure of Geometric Number Sequences.” Notes on Number Theory and Discrete Mathematics 17, no. 3 (2011): 31-37.

MLA

Leyendekkers, JV, and AG Shannon. “The Structure of Geometric Number Sequences.” Notes on Number Theory and Discrete Mathematics 17.3 (2011): 31-37. Print.

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