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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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 Z4 fall in even rows, R1, in Class ̅14 with R1 = 6Kj and Kj = ½ n(3n ± 1) (depending on the parity of j). The n values are found to equal the rows that the primes occupy when Z6 is set as a tabular array. The primes in Z4 equal a unique sum or difference of squares (x2 ± y2) and via n, the (x, y) pairs can be identified within the Z6 structure. The n values for composites follow well-defined linear functions that permit easy sorting. Finally, the parameter n in the well-known function P = h2n − 1 has been identified as the row in one or more of the Modular Rings Z10, Z6 or Z4 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 Z4 and in Class ̅14. 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 Z4 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 Z6. Notes on Number Theory & Discrete Mathematics. 7(4) 2001: 115-124.
- J. V. Leyendekkers & A. G. Shannon. An Analysis of Twin Primes h2n −1 Using Modular Rings Z6 and Z4. 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 Z6. 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 Z6
- The characteristics of primes and other integers within the modular ring Z4 and in class ̅3
- The characteristics of primes and other integers within the modular ring Z4 and in class ̅1
- 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 h2n−1 using modular rings ℤ6 and ℤ4
- 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.