The structure of even powers in Z3: Critical structural factors that prevent the formation of even—powered triples greater than squares

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 17, 2011, Number 3, Pages 26—30
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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 illustrates the critical structural factors which underpin the failure of (N4m + M4m) ever to equal an equivalent power. The number 3 plays a vital role as integers divisible by 3, when raised to an even power of the form 4m, have rows in a table of modular rings which are triangular numbers, whereas other integers raised to the same power have rows which are pentagonal numbers. The substructure within these sequences of pentagonal numbers is order within order, analogous to structure in chaos theory.

Keywords

  • Integer structure analysis
  • Modular rings
  • Pentagonal numbers
  • Triangular numbers

AMS Classification

  • 11A41
  • 11A07

References

  1. Euler, L. De mirabilis proprietatibus numerorum pentagonalium. Acta Academiae Scientarum Imperialis Petropolitinae. Vol. 4, 1783, No. 1, 56–75.
  2. Leyendekkers, J. V., A. G. Shannon, J. M. Rybak. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph 2007, No. 9.
  3. Leyendekkers, J. V., A. G. Shannon, C. K. Wong. 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.
  4. Leyendekkers, J. V., A. G. Shannon. The Integers Structure of the Difference of Two Odd Integers Raised to an Even Power. Notes on Number Theory and Discrete Mathematics. Vol. 16, 2010, No. 1, 1–4.
  5. 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.
  6. Leyendekkers, J. V., A. G. Shannon. Why Are Some Right-end Digits Integers Absent in Primitive Pythagorean Triples? Notes on Number Theory and Discrete Mathematics. Vol. 17, 2011, No. 1, 37–44.
  7. Ono, K., N. Robbins, B. Wilson. Some Recurrences for Arithmetical Functions. Journal of the Indian Mathematical Society. Vol. 62, 1996, 29–50.

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Cite this paper

APA

Leyendekkers, J., & Shannon, A. (2011). 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, 17(3), 26-30.

Chicago

Leyendekkers, JV, and AG 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 17, no. 3 (2011): 26-30.

MLA

Leyendekkers, JV, and AG 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 17.3 (2011): 26-30. Print.

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