Asymptotic formulae for the number of repeating prime sequences less than N

Christopher L. Garvie
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 22, 2016, Number 4, Pages 29–40
Full paper (PDF, 233 Kb)

Details

Authors and affiliations

Christopher L. Garvie
Texas Natural Science Center, University of Texas at Austin
10100 Burnet Road, Austin, Texas 78758, USA

Abstract

It is shown that prime sequences of arbitrary length, of which the prime pairs, (p, p+2), the prime triplet conjecture, (p, p+2, p+6) are simple examples, are true and that prime sequences of arbitrary length can be found and shown to repeat indefinitely. Asymptotic formulae comparable to the prime number theorem are derived for arbitrary length sequences. An elementary proof is also derived for the prime number theorem and Dirichlet’s Theorem on the arithmetic progression of primes.

Keywords

• Primes
• Sequences
• Distribution

AMS Classification

• 11A41

References

1. Everest, G., & Ward, T. (2005) An Introduction to Number Theory. Springer-Verlag, London.
2. Hardy, G.H., & Wright, E.M. (1959) An Introduction to the Theory of Numbers. Oxford University Press, London.
3. Oliveira e Silva, T. (2008). “Tables of values of pi(x) and of pi2(x)”, Aveiro University. http://sweet.ua.pt/tos/primes.html (retrieved 7 January 2011).