On a Recursive Formula for the Sequence of Primes and Applications to the Twin Prime Problem

In this paper we give a recursive formula for the sequence of primes {pn} and apply it to find a necessary and sufficient condition in order that a prime number pn+1 is equal to pn+2. Applications of previous results are given to evaluate the probability that pn+1 is of the form pn + 2; moreover we prove that the limit of this probability is equal to zero as n goes to ∞. Finally, for every prime pn we construct a sequence whose terms that are in the interval [pn − 2, pn+1 − 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. Introduction. It is well known there are many open problems about the sequence of primes (see [1], [3], [4], [5], [6], [7]); one of these is the twin prime problem, which consists in finding out if there exist infinitely many primes p such that p+2 is also prime (if the numbers p and p+2 are both primes, they are called twin primes). In the first part of this paper we give a recursive formula for the sequence of primes {pn}, that we think be novel (for other recursive formulas see reference A 17 p. 37 of [3]). In the second part we apply it to find a necessary and sufficient condition in order that a prime number pn+1 is equal to pn+2. Moreover applications of previous results are given to evaluate the probability that pn+1 is of the form pn + 2 and from this we deduce that the limit of this probability is equal to zero when n goes to ∞. Finally, in the third part, for every prime pn we construct a sequence Σpn whose terms that are in the interval [p 2 n−2, pn+1−2[ are the first terms of two twin primes and moreover we prove a theorem on the mean number of the terms of the sequence Σpn that are in the interval [p 2 n−2, pn+1−2[, which makes furthermore plausible that the set of twin primes is infinite. In the sequel we put as usual —————————– Mathematics Subject Classifications: 11A41, 11B25, 40A05, 40A20.