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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## Details

### 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 (*N*^{4m} + *M*^{4m}) 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 4*m*, 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

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- Leyendekkers, J. V., A. G. Shannon, J. M. Rybak. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph 2007, No. 9.
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- 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.
- 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.
- 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.
- Ono, K., N. Robbins, B. Wilson. Some Recurrences for Arithmetical Functions. Journal of the Indian Mathematical Society. Vol. 62, 1996, 29–50.

## Related papers

## Cite this paper

APALeyendekkers, J., & Shannon, A. (2011). The structure of even powers in *Z*_{3}: 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.

Leyendekkers, JV, and AG Shannon. “The Structure of Even Powers in *Z*_{3}: 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.

Leyendekkers, JV, and AG Shannon. “The Structure of Even Powers in *Z*_{3}: 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.