# An Analytic Approximation to the Density of Twin Primes

### Abstract

The highly irregular and rough fluctuations of the twin primes below or equal to a positive integer *x * are considered in this study. The occurrence of a twin prime on an interval [0,x] is assumed to be random. In particular, we considered the waiting time between arrivals of twin primes as approximated by a geometric distribution which possesses the discrete memory-less property. For large n, the geometric distribution is well-approximated by the exponential distribution. The number of twin primes less or equal to x will then follow the Poisson distribution with the same rate parameter as the exponential distribution. The results are compared with the Hardy-Littlewood conjecture on the frequency of twin primes. We successfully demonstrated that for large n, the proposed model is superior to the H-L conjecture in predicting the frequency of twin primes.

### References

Banks,W.D., Freiberg,T.,& Maynard, J. (2014). On the limit points of the sequence of normalized prime gaps. arXiv preprint arXiv:1404.5094

Hardy, G. H., & Littlewood, J. E. (1984). SOME PROBLEMS OF’PARTITIO NUMERORUM’; III: ON THE EXPRESSION OF A NUMBER AS A SUM OF PRIMES. In Goldbach Conjecture(pp. 21-60).

Fliegel, Henry F.; Robertson, Douglas S. (1989). "Goldbach's Comet: the numbers related to Goldbach's Conjecture". Journal of Recreational Mathematics. 21 (1): 1–7.

Ford,K., Green,B.,Konyagin,S., & Tao, T.(2014).Large gaps between consecutive prime numbers. arXiv preprint arXiv: 1408.4505

Maynard,J.(2013). Small gaps between primes. arXiv preprint arXiv:1311.4600

Maynard, J. (2013). Bounded length intervals containing two primes and an almost-prime. Bulletin of the London Mathematical Society, 45(4), 753-764.

Padua, R. N., & Libao, M. F. (2017). ON STOCHASTIC APPROXIMATIONS TO THE DISTRIBUTION OF PRIMES AND PRIME METRICS. Journal of Higher Education Research Disciplines, 1(1), 33-38.

Regalado, D. & Azura, R. On a Mixed Regression Estimator for the Density of Prime Gaps. Journal of Higher Education Research Disciplines. 1 (2), 48-57.

Sloane, N.J.A. (ed.). "Sequence A001692 (Number of irreducible polynomials of degree n over GF(5); dimensions of free Lie algebras)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2011-02-02. -- A page of number theoretical constants

Young, J., & Potler, A. (1989). First occurrence prime gaps. Mathematics of Computation, 221-224.

*Recoletos Multidisciplinary Research Journal*,

*6*(2). https://doi.org/10.32871/rmrj1806.02.05

Copyright of the Journal belongs to the University of San Jose-Recoletos