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

Fiorito, Giovanni

Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 3, pp. 667-680

Résumé

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.