# 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.

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