Equations for primes obtained from integer structure

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
Full paper (PDF, 224 Kb)

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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

Abstract

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 = 2^2q(12m(3m ± 1) + 1) or 2^2q(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.

Keywords

• Primes
• Composites
• Modular rings
• Right-end digits
• Integer structure
• Triangular numbers
• Pentagonal numbers

AMS Classification

• 11A41
• 11A07

References

1. Leyendekkers, J.V., A.G. Shannon, J.M. Rybak. 2007. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No 9.
2. 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.
3. 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.