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)**

## Details

### 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 4*R*_{1} + 1 are equal to a sum of squares. Such primes are in Class ̅1_{4} ⊂ Z_{4}, a modular ring. The rows of squares in Z_{4} are well defined and this permits equalities for the primes to be derived from the integer structure. These equalities have the forms *R*_{1} = *r*_{1} + *r*_{0} where *r*_{1} = 3*n*(3*n* ± 1) or 2 + 9n′(n′ + 1) and *r*_{0} = 2^{2q}(12*m*(3*m* ± 1) + 1) or 2^{2q}(4(2 + 9*m*′(*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 ̅3_{4} may be obtained in the same manner using *p* = *y*^{2} − *x*^{2} with (*y* −* x*) = 1.

### Keywords

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

### AMS Classification

- 11A41
- 11A07

### References

- 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 Z
_{6}.*Notes on Number Theory and Discrete Mathematics.*14(4): 10-15.

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## 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.