# On a recursive formula for the sequence of primes and applications to the twin prime problem

Bollettino dell'Unione Matematica Italiana (2006)

- Volume: 9-B, Issue: 3, page 667-680
- ISSN: 0392-4041

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topFiorito, Giovanni. "On a recursive formula for the sequence of primes and applications to the twin prime problem." Bollettino dell'Unione Matematica Italiana 9-B.3 (2006): 667-680. <http://eudml.org/doc/289622>.

@article{Fiorito2006,

abstract = {In this paper we give a recursive formula for the sequence of primes $\\{p_n\\}$ and apply it to find a necessary and sufficient condition in order that a prime number $p_\{n+1\}$ is equal to $p_\{n\}+2$. Applications of previous results are given to evaluate the probability that $p_\{n+1\}$ is of the form $p_\{n\}+2$; moreover we prove that the limit of this probability is equal to zero as $n$ goes to $\infty$. Finally, for every prime $p_n$ we construct a sequence whose terms that are in the interval $[p_n^2 - 2 , p_\{n+1\}^2-2[$ are the first terms of two twin primes. This result and some of its implications make furthermore plausible that the set of twin primes is infinite.},

author = {Fiorito, Giovanni},

journal = {Bollettino dell'Unione Matematica Italiana},

language = {eng},

month = {10},

number = {3},

pages = {667-680},

publisher = {Unione Matematica Italiana},

title = {On a recursive formula for the sequence of primes and applications to the twin prime problem},

url = {http://eudml.org/doc/289622},

volume = {9-B},

year = {2006},

}

TY - JOUR

AU - Fiorito, Giovanni

TI - On a recursive formula for the sequence of primes and applications to the twin prime problem

JO - Bollettino dell'Unione Matematica Italiana

DA - 2006/10//

PB - Unione Matematica Italiana

VL - 9-B

IS - 3

SP - 667

EP - 680

AB - In this paper we give a recursive formula for the sequence of primes $\{p_n\}$ and apply it to find a necessary and sufficient condition in order that a prime number $p_{n+1}$ is equal to $p_{n}+2$. Applications of previous results are given to evaluate the probability that $p_{n+1}$ is of the form $p_{n}+2$; moreover we prove that the limit of this probability is equal to zero as $n$ goes to $\infty$. Finally, for every prime $p_n$ we construct a sequence whose terms that are in the interval $[p_n^2 - 2 , p_{n+1}^2-2[$ are the first terms of two twin primes. This result and some of its implications make furthermore plausible that the set of twin primes is infinite.

LA - eng

UR - http://eudml.org/doc/289622

ER -

## References

top- CRANDALL, R. - POMERANCE, C., Prime Numbers A Computational Perspective, Springer-VerlagNew York (2001). Zbl0995.11072
- FIORITO, G., On Properties of Periodically Monotone Sequences, Applied Mathematics and Computation, 72 (1995), 259-275. Zbl0838.40002
- GUY, R. K., Unsolved problems in Number Theory, Springer-VerlagNew York (1994). Zbl0805.11001
- HARDY, G. - WRIGHT, E., An Introduction to the Theory of Numbers, Clarendon PressOxford (1954). Zbl0058.03301
- IRELAND, K. - ROSEN, M., A Classical Introduction to Modern Number Theory, Springer-VerlagNew York (1981). Zbl0712.11001
- MURTY, M. R., Problems in Analytic Number Theory, Springer-Verlag (1999). Zbl0911.11001
- NATHANSON, M. B., Elementary Methods in Number Theory, Springer-Verlag (1999).

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