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

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

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## Abstract

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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.

## How to cite

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Fiorito, 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

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

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