An analysis of twin primes h2n−1 using modular rings ℤ6 and ℤ4

J. Leyendekkers and A. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 7, 2001, Number 1, Pages 21–28
Full paper (PDF, 354 Kb)

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Authors and affiliations

J. Leyendekkers
The University of Sydney, 2006, Australia

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

Abstract

Twin primes of the form h2n−1 are analysed within the modular ring ℤ6. The values of h are odd and 3|h for the lowest valued twin prime, p2. The other values of h fall in either the second (26) or fourth (46) class of ℤ6, depending on the parity of n. Functional relationships are developed for the various h, n and rows within ℤ6. All p2 fall in class 26 and the larger prime of the twin pair, p4, always falls in class 46. With n2 = 1 and n4 > 1, p2 falls in class one (14) of the modular ring ℤ4 and hence equals a unique set of squares (x2 + y2). Analysis of the distribution of the (x,y) pair reveals an interesting prime sequence related to the Fibonacci sequence.

AMS Classification

  • 11A51
  • 11B39

References

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Related papers

  1. 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. & Shannon, A. (2001). An analyses of twin primes h2n−1 using modular rings ℤ6 and ℤ4. Notes on Number Theory and Discrete Mathematics, 7(1), 21-28.

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