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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## Details

### 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 *h*2^{n}−1 are analysed within the modular ring ℤ_{6}. The values of *h* are odd and 3|*h* for the lowest valued twin prime, *p*_{2}. The other values of *h* fall in either the second (2_{6}) or fourth (4_{6}) class of ℤ_{6}, depending on the parity of *n*. Functional relationships are developed for the various *h*, *n* and rows within ℤ_{6}. All *p*_{2} fall in class 2_{6} and the larger prime of the twin pair, *p*_{4}, always falls in class 4_{6}. With *n*_{2} = 1 and *n*_{4} > 1, *p*_{2} falls in class one (1_{4}) of the modular ring ℤ_{4} and hence equals a unique set of squares (*x*^{2} + *y*^{2}). 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

- 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 *h*2* ^{n}*−1 using modular rings ℤ

_{6}and ℤ

_{4}. Notes on Number Theory and Discrete Mathematics, 7(1), 21-28.