Figurate numbers in the modular ring Z₄

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 22, 2016, Number 4, Pages 49–55
Full paper (PDF, 164 Kb)

Details

Authors and affiliations

J. V. Leyendekkers
Faculty of Science, The University of Sydney, NSW 2006, Australia

A. G. Shannon
Emeritus Professor, University of Technology Sydney, NSW 2007, Australia
Campion College, PO Box 3052, Toongabbie East, NSW 2146, Australia

Abstract

The sums of odd integers in classes ̅1₄, ̅3₄ ⊂ Z₄, a modular ring, show clear distinctions between the two classes. In particular, the sum for class ̅1₄ is related to the Golden Ratio family of sequences, and in this class when the position of an odd integer is a prime number, then the sum always has a factor of 6. Sums of the primes in these classes can be primes but the structures are quite different, and no sums of odd integers in general are primes. The sums are related to the sequences of triangular numbers and hexagonal numbers.

Keywords

• Prime numbers
• Composite numbers
• Right-end-digits
• Modular rings
• Generalized Golden Ratio
• Generalized Fibonacci numbers
• Triangular numbers
• Hexagonal numbers

AMS Classification

• 11B39
• 11B50

References

1. Atanassov, K. T., Atanassova, V., Shannon, A. G., & Turner, J. C. (2002) New Visual Perspectives on Fibonacci Numbers. New Jersey/Singapore: World Scientific.
2. Cook, C. K., & Bacon, M. R. (2014) Some polygonal number summation formulas. The Fibonacci Quarterly. 52(4), 336–343.
3. Deza, E., & Deza, M. M. (2012) Figurate Numbers. New York/Singapore: World Scientific.
4. Leyendekkers, J. V., & Shannon, A. G. (2015) The Golden Ratio Family and Generalized Fibonacci Numbers. Journal of Advances in Mathematics. 10(1), 3130–3137.
5. Leyendekkers, J. V., & Shannon, A. G. (2015) The Odd-number Sequence: Squares and Sums. International Journal of Mathematical Education in Science & Technology. DOI: 10.1080/0020739X.2015.1044042.
6. Leyendekkers, J. V., & Shannon, A. G. (2015) The Sum of Squares for Primes. Notes on Number Theory and Discrete Mathematics.  21(4), 17–21.
7. Shannon, A.G., Anderson, P.G., & Horadam, A.F. (2006) Properties of Cordonnier, Perrin and Van der Laan Numbers. International Journal of Mathematical Education in Science & Technology, 37(7), 825–831.
8. Shannon, A.G., Cook, C. K., & Hillman, R. A. (2013) Some Aspects of Fibonacci Polynomial Congruences. Annales Mathematicae et Informaticae. 41, 211–217.