The characteristics of primes and other integers within the modular ring \bm{Z_4} and in class \overline{1}

J. V. Leyendekkers, J. M. Rybak and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 4, 1998, Number 1, Pages 1–17
Full paper (PDF, 663 Kb)

Details

Authors and affiliations

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

J. M. Rybak
The University of Sydney, 2006, Australia

A. G. Shannon
University of Technology, Sydney, 2007, Australia

Abstract

The integer structure of Class \overline{1} in the modular ring Z_4 has been analysed in detail. Most integers of this category equal a sum of two squares (x^2 + y^2). Those that do not are non-primes. The primes are distinguished by having a unique \langle x,y \rangle  pair that has no common factors. Other integers in Class \overline{1} have multiple values of \langle x,y \rangle  or more rarely a single \langle x,y \rangle pair with common factors. Methods of estimating \langle x,y \rangle  pairs are given. These are based on the class structure within Z_4 and the right-most end digit characteristics. The identification of primes is consequently facilitated.

References

  1. J.V. Leyendekkers, J.M. Rybak and A.G. Shannon, Analysis of Diophantine Properties using Modular Rings with Four and Six Classes. Notes on Number Theory and Discrete Mathematics 3, 2, 1997, 61-74.
  2. J.V. Leyendekkers, J.M. Rybak, The generation and analysis of Pythagorean triples within a two-parameter grid. International Journal of Mathematical Education in Science and Technology, 26,(6), 1995: 787-793.
  3. Richard E. Crandall, The challenge of large numbers, Scientific American, 276(2),1997:58-62.

Related papers

  1. Leyendekkers, J. V., Rybak, J. M., & Shannon, A. G. (1997). Analysis of Diophantine Properties using Modular Rings with Four and Six Classes. Notes on Number Theory and Discrete Mathematics, 3(2), 61-74.
  2. Leyendekkers, J. V., & Shannon, A. G. (2003). Some characteristics of primes within modular rings. Notes on Number Theory and Discrete Mathematics, 9(3), 49-58.

Cite this paper

Leyendekkers, J. V., Rybak, J. M. & Shannon, A. G. (1998). The characteristics of primes and other integers within the modular ring Z_4 and in class \overline{1}. Notes on Number Theory and Discrete Mathematics, 4(1), 1-17.

Comments are closed.