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
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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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