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)

Details

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

Abstract

Simple functions were obtained for the rows of odd powers in the modular ring Z₄, wherein integer N is represented by N = 4r_i + 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…

Keywords

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

AMS Classification

• 11A41
• 11A07

References

1. Beiler, Arthur. 1966. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, p.115.
2. Dickson, Leonard E. 1952. History of the Theory of Numbers, New York: Chelsea.
3. Lehmer, D. N. 1900. Asymptotic Evaluation of Certain Totient Sums. American Journal of Mathematics. 22: 293-335.
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.