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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## Details

### 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 Z_{4} fall in even rows, *R*_{1}, in Class ̅1_{4} with *R*_{1} = 6*K _{j}* and

*K*= ½

_{j}*n*(3

*n*± 1) (depending on the parity of

*j*). The

*n*values are found to equal the rows that the primes occupy when Z

_{6}is set as a tabular array. The primes in Z

_{4}equal a unique sum or difference of squares (

*x*

^{2}±

*y*

^{2}) and via

*n*, the (

*x*,

*y*) pairs can be identified within the Z

_{6}structure. The

*n*values for composites follow well-defined linear functions that permit easy sorting. Finally, the parameter

__in the well-known function__

*n**=*

__P__*h*2

*− 1 has been identified as the row in one or more of the Modular Rings Z*

^{n}_{10}, Z

_{6}or Z

_{4}that contains one or more primes.

### AMS Classification

- 11A07
- 11A41

### References

- J. V. Leyendekkers, J.M. Rybak & A. G. Shannon. The Characteristics of Primes and Other Integers within the Modular Ring Z
_{4}and in Class ̅1_{4}. Notes on Number Theory & Discrete Mathematics. 4(1) 1998: 1-17. - J. V. Leyendekkers, J.M. Rybak & A. G. Shannon. The Characteristics of Primes and Other Integers within the Modular Ring Z
_{4}and in class ̅3. Notes on Number Theory & Discrete Mathematics. 4(1) 1998: 18-37. - J. V. Leyendekkers & A. G. Shannon. An Analysis of Mersenne-Fibonacci and Mersenne-Lucas Primes. Notes on Number Theory & Discrete Mathematics. 5(1) 1999: 1-26.
- J. V. Leyendekkers & A. G. Shannon. The Goldbach Conjecture Primes within a Modular Ring. Notes on Number Theory & Discrete Mathematics. 6(4) 2000: 101-112.
- J. V. Leyendekkers & A. G. Shannon. Twin Primes and the Modular Ring Z
_{6}. Notes on Number Theory & Discrete Mathematics. 7(4) 2001: 115-124. - J. V. Leyendekkers & A. G. Shannon. An Analysis of Twin Primes
*h*2^{n}−1 Using Modular Rings Z_{6}and Z_{4}. Notes on Number Theory & Discrete Mathematics. 7(1) 2001:21-28. - J. V. Leyendekkers & A. G. Shannon. A Note on Twin Primes and the Modular Ring Z
_{6}. International Journal of Mathematical Education in Science & Technology. 33(2) 2002: 303-306. - J. V. Leyendekkers & A. G. Shannon. Powers as a Difference of Squares: The Effect on Triples. Notes on Number Theory & Discrete Mathematics. 8(3) 2002: 95-106.
- J. V. Leyendekkers, J. M. Rybak & A. G. Shannon. Integer Class Properties Associates with an Integer Matrix. Notes on Number Theory & Discrete Mathematics. 1(2) 1995: 53-59.
- J. V. Leyendekkers, J. M. Rybak & A. G. Shannon. Analysis of Diophantine Properties Using Modular Rings with Four and Six Classes. Notes on Number Theory & Discrete Mathematics. 3(2) 1997: 61-74.
- Hans Riese Prime Numbers and Computer Methods for Factorization. Progress in Mathematics, Volume 126. Boston: Birkhauser, 1994.
- H. Beiler. Recreations in the Theory of Numbers. New York: Dover, 1964.

## Related papers

- The analysis of twin primes within Z
_{6} - The characteristics of primes and other integers within the modular ring
*Z*and in class ̅3_{4} - The characteristics of primes and other integers within the modular ring
*Z*and in class ̅1_{4} - The Goldberg-conjecture primes within a modular ring
- An analysis of Mersenne—Fibonacci and Mersenne—Lucas primes
- Analysis of Diophantine properties using modular rings with four and six classes
- Powers as a difference of squares: The effect on triples
- An analysis of twin primes
*h*2^{n}−1 using modular rings ℤ_{6}and ℤ_{4} - Integer class properties associated with an integer matrix

## 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.