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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## 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

Even and odd integers of the form *N ^{m}* in the modular ring

*Z*

_{4}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

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- Hardy, G. H., E. M. Wright. An Introduction to the Theory of Numbers, 3rd edition. London: Oxford University Press, 1954.
- Leyendekkers, J. V., A. G. Shannon, J. M. Rybak. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No. 9, 2007.
- 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.
- 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.
- 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.
- Leyendekkers, J. V., A. G. 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. Vol. 17, 2011, No. 3, 26–30. - Leyendekkers, J. V., A. G. Shannon. The Structure of π. Notes on Number Theory and Discrete Mathematics. Vol. 17, 2011, No. 4, 61–68.
- Leyendekkers, J. V., A. G. Shannon. The Modular Ring
*Z*_{5}. Notes on Number Theory and Discrete Mathematics, Vol. 18, 2012, No. 2, 28–33.

## Related papers

## Cite this paper

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

ChicagoLeyendekkers, 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.

MLALeyendekkers, 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.