J. V. Leyendekkers and A. G. Shannon

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 17, 2011, Number 2, Pages 47—51

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## Details

### Authors and affiliations

J. V. Leyendekkers

*The University of Sydney, 2006, Australia*

A. G. Shannon

*Faculty of Engineering & IT, University of Technology Sydney
2007 Australia*

### Abstract

The characteristics within modular rings of the integer three are discussed. This integer has unique row structures in modular rings which appear to underlie restraints on various aspects of triples, particularly the factors and powers structure of the components. The function *N* = *x ^{m}* + 2

*, with*

^{n}*m*even and

*n*odd but

*x*not divisible by 3, always has 3 as a factor, and a majority of elements of the sequence of triangular numbers {

*N*} are such that 3|

_{T}*N*. The modular ring Z

_{T}_{3}and the distribution of primes within its structure are also discussed.

### Keywords

- Integer structure analysis
- Modular rings
- Prime numbers
- Triangular numbers
- Pentagonal numbers
- Octagonal numbers
- Repunits

### AMS Classification

- 11A41
- 11A07

### References

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*Mathematical Gems II*. Washington, DC: Mathematical Association of America, pp. 98-104. - Leyendekkers, J.V., A.G. Shannon. 1999. Analyses of Row Expansions within the Octic ‘Chess’ Modular Ring
*Z*_{8}.*Notes on Number Theory & Discrete Mathematics*. 5(3): 102-114. - Leyendekkers, J.V., A.G. Shannon, J.M. Rybak. 2007.
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*International Journal of Mathematical Education in Science & Technology.*42: 102-105. - Leyendekkers, J.V., A.G. Shannon. 2010. Integer Structure Analysis of Odd Powered Triples: The Significance of Triangular versus Pentagonal Numbers.
*Notes on Number Theory & Discrete Mathematics*. 16 (4): 6-13. - 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. - Leyendekkers, J.V., A.G. Shannon. 2011. Structure Analysis of the Perimeters of Primitive Pythagorean Triples.
*Notes on Number Theory & Discrete Mathematics*In press. - Sloane, N.J.A., Simon Plouffe. 1995.
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*Journal of Recreational Mathematics.*2: 139-146.

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## Cite this paper

APALeyendekkers, J. V., and Shannon, A. G. (2011). Modular rings and the integer 3. Notes on Number Theory and Discrete Mathematics, 17(2), 47-51.

ChicagoLeyendekkers, JV, and AG Shannon. “Modular Rings and the Integer 3.” Notes on Number Theory and Discrete Mathematics 17, no. 2 (2011): 47-51.

MLALeyendekkers, JV, and AG Shannon. “Modular Rings and the Integer 3.” Notes on Number Theory and Discrete Mathematics 17.2 (2011): 47-51. Print.