J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 16, 2010, Number 3, Pages 1—10
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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
PO Box 123, Kensington, NSW 1465, Australia
Integer structure illustrates how primes represented by 4R1 + 1 are equal to a sum of squares. Such primes are in Class ̅14 ⊂ Z4, a modular ring. The rows of squares in Z4 are well defined and this permits equalities for the primes to be derived from the integer structure. These equalities have the forms R1 = r1 + r0 where r1 = 3n(3n ± 1) or 2 + 9n′(n′ + 1) and r0 = 22q(12m(3m ± 1) + 1) or 22q(4(2 + 9m′(m′ + 1)) + 1), with q = 0, 1, 2, 3,… and n, m yielding the pentagonal numbers, and n′, m′ the triangular numbers. When n = m the equations are similar to Euler’s prime equation. Equations for the remaining primes, in Class ̅34 may be obtained in the same manner using p = y2 − x2 with (y − x) = 1.
- Modular rings
- Right-end digits
- Integer structure
- Triangular numbers
- Pentagonal numbers
- Leyendekkers, J.V., A.G. Shannon, J.M. Rybak. 2007. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No 9.
- Leyendekkers, J.V., A.G. Shannon. 2008. Analysis of Primes Using Right-end-digits and Integer Structure. Notes on Number Theory and Discrete Mathematics. 14(3): 1-10.
- Leyendekkers, J.V., A.G. Shannon. 2008. The identification of Rows of Primes in the Modular Ring Z6. Notes on Number Theory and Discrete Mathematics. 14(4): 10-15.
Cite this paper
Leyendekkers, J. V., and Shannon, A. G. (2010). Equations for primes obtained from integer structure. Notes on Number Theory and Discrete Mathematics, 16(3), 1-10.