Some characteristics of primes within modular rings

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 9, 2003, Number 3, Pages 49—58
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Authors and affiliations

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

A. G. Shannon
Warrane College, The University of New South Wales, 1465,
& KvB Institute of Technology, North Sydney, 2060, Australia

Abstract

The squares of primes in the Modular Ring Z4 fall in even rows, R1, in Class ̅14 with R1 = 6Kj and Kj = ½ n(3n ± 1) (depending on the parity of j). The n values are found to equal the rows that the primes occupy when Z6 is set as a tabular array. The primes in Z4 equal a unique sum or difference of squares (x2 ± y2) and via n, the (x, y) pairs can be identified within the Z6 structure. The n values for composites follow well-defined linear functions that permit easy sorting. Finally, the parameter n in the well-known function P = h2n − 1 has been identified as the row in one or more of the Modular Rings Z10, Z6 or Z4 that contains one or more primes.

AMS Classification

  • 11A07
  • 11A41

References

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  2. J. V. Leyendekkers, J.M. Rybak & A. G. Shannon. The Characteristics of Primes and Other Integers within the Modular Ring Z4 and in class ̅3. Notes on Number Theory & Discrete Mathematics. 4(1) 1998: 18-37.
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Cite this paper

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.

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