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

Giovanni Fiorito

Bollettino dell'Unione Matematica Italiana (2006)

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

Abstract

top
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 ∞ . 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

top

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

top
  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). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.