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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 $\mathrm{\infty }$. 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...