J. V. Leyendekkers and A. G. Shannon

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 10, 2004, Number 3, Pages 77—83

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## Details

### Authors and affiliations

J. V. Leyendekkers

*The University of Sydney, 2006 Australia*

A. G. Shannon

*Warrane College, Kensington, NSW 1465,
& KvB Institute of Technology, North Sydney, NSW 2060, Australia *

### Abstract

With the exception of 2 and 3, primes are only found in two classes of the modular ring Z_{6}. The rows of the tabular display of this ring which contain composites in these two classes are given by *R* = *R*_{0}+ *pt*, *t* = 0, 1, 2, 3, …, in which *R*_{0} is a function of *p* that is class specific. The number of composites, *n _{c}*, in the two classes can be calculated from:

(*a* = 1 or 2 depending on the class of *p*),

where *p _{L}* is the prime less than and closest to √

*N*. The

*Q*terms are quantities which arise from the characteristics of the factors of the composites. Subtraction of

_{i}*n*from the total integers in each class yields the number of primes for that class, and hence the total number of primes in the interval.

_{c}### AMS Classification

- 11A41
- 11A07

### References

- J.V. Leyendekkers & A.G. Shannon, The Analysis of Twin Primes within Z
_{6}. Notes on Number Theory & Discrete Mathematics. 7(4) 2001: 115-124. - 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 & A.G. Shannon, Some Characteristics of Primes within Modular Rings. Notes on Number Theory & Discrete Mathematics. 9(3) 2003:49-58.
- J.V. Leyendekkers & A.G. Shannon, Analysis of Quadratic Diophantine Equations with Fibonacci-number Solutions. International Journal of Mathematical Education in Science & Technology. In press.
- J.V. Leyendekkers & A.G. Shannon, Structure of Fibonacci Numbers within Modular Rings. Notes on Number Theory & Discrete Mathematics. In press.
- Karl, Sabbagh, Dr Riemann’s Zeros. London: Atlantic Books, 2002.

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## Cite this paper

Leyendekkers, J. V., and Shannon, A. G. (2004). Using integer structure to calculate the number of primes in a given interval. Notes on Number Theory and Discrete Mathematics, 10(3), 77-83.