Rows of odd powers in the modular ring Z4

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 16, 2010, Number 2, Pages 29–32
Full paper (PDF, 37 Kb)


Authors and affiliations

J. V. Leyendekkers
The University of Sydney, 2006, Australia

A. G. Shannon
Warrane College, The University of New South Wales
PO Box 123, Kensington, NSW 1465, Australia


Simple functions were obtained for the rows of odd powers in the modular ring Z4, wherein integer N is represented by N = 4ri + i, i = 0, 1, 2, 3. These row functions are based on the row functions for squares. When 3 ∤ N , the row of N2 = 3n(3n ±1) , or when 3|N, the row of N2 = 2 + 18(½n(n+1)), n = 0, 1, 2, 3…


  • Primitive Pythagorean triples
  • Modular rings
  • Triangular numbers
  • Pentagonal numbers

AMS Classification

  • 11A41
  • 11A07


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  4. Leyendekkers, J.V., A.G. Shannon, J.M. Rybak. 2007. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No.9.
  5. Leyendekkers, J.V., A.G. Shannon. 2010. Equations for Primes Obtained from Integer Structure. Notes on Number Theory and Discrete Mathematics. Submitted.
  6. Mohapatra, Amar Kumar, Nupur Prakash. 2010. Generalized Formula to Determine Pythagorean Triples. International Journal of Mathematical Education in Science and Technology. 41: 131-135.

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Cite this paper

Leyendekkers, J. V., & Shannon, A. G. (2010). Rows of odd powers in the modular ring Z4. Notes on Number Theory and Discrete Mathematics, 16(2), 29-32.

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