J. V. Leyendekkers and A. G. Shannon

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 7, 2001, Number 4, Pages 115—124

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## Details

### Authors and affiliations

J. V. Leyendekkers

*The University of Sydney
NSW 2006, Australia*

A. G. Shannon

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

### Abstract

The modular ring Z_{6} defines integers via (6*r*** _{i}** + (

*i*– 3)) where i is the Class and r, the row when tabulated in an array. Since only Classes 2

**and 4**

_{6}**(; contain odd primes, this modular ring is ideally suited for the analysis of twin primes. In considering a series of integers, a simple method is used to calculate rows (F rows) that do not contain twin primes. This allows the distribution of other primes to be found. Then, in considering the corresponding array of rows, elimination of the**

_{6}*F*rows yields the rows which contain twin primes. The calculations are facilitated by the use of the right-end digit (RED) technique.

### AMS Classification

- 11A41
- 11A51

### References

- Abramowitz, M. & Stegun, I. A. 1964. Handbook of Mathematical Functions.

Washington, DC: National Bureau of Standards. - Boyer, Carl B. 1968. A History of Mathematics. Princeton: Princeton University Press.
- Dunham, William, 1994. The Mathematical Universe. New York: John Wiley.

Forbes, T. 1997. A Large Pair of Twin Primes. Mathematics of Computation. 66(217): 451-455. - Halberstam, H. & Richert, H-E. 1974. Sieve Methods. New York: Academic Press.
- Holben, C.A. & Jordan, J.H. 1968. The Twin Prime Problem and Goldbach’s Conjecture in the Gaussian Integers. The Fibonacci Quarterly. 6(5): 81-85,92.
- Leyendekkers, J..V., Rybak, J.M., & Shannon, A.G. 1995. Integer Class Properties Associated with an Integer Matrix. Notes on Number Theory and Discrete Mathematics. 1(2): 53-59.
- Leyendekkers, J.V. & Rybak, J.M. 1995. The Generation and Analysis of Pythagorean Triples Within a Two-parameter Grid. International Journal of Mathematical Education in Science & Technology. 26(6): 787-793.
- 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.
- Leyendekkers, J..V., & Shannon, A.G. 2000. The Goldberg Conjecture Primes within a Modular Ring. Notes on Number Theory and Discrete Mathematics. 6(4): 101-112..
- Ramachandra, K. 1998. Many Famous Conjectures on Primes; Meagre But Precious Progress of a Deep Nature. Mathematics Student. 67(1-4): 187-199.
- Ribenboim, P. 1989. The Book of Prime Number Records. 2nd edition. New York: Springer.
- Riesel, Hans. 1994. Prime Numbers and Computer Methods for Factorization. Progress in Mathematics, Volume 126. Boston: Birkhauser.

## Related papers

- Leyendekkers, J. V., & Shannon, A. G. (2018). An indicator characteristic for twin prime formation independent of integer size. Notes on Number Theory and Discrete Mathematics, 24(1), 10-15.
- 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). The analysis of twin primes within Z_{6}. Notes on Number Theory and Discrete Mathematics, 7(4), 115-124.