## Can the Twin Prime Conjecture be Proven

**Authors:** James Edwin Rock

Let P_n be the n_th prime. For twin primes P_n – P_(n-1) = 2. Let X be the number of (6j –1, 6j+1) pairs in the interval [P_n, P_n^2]. The number of twin primes (TPAn) in [P_n, P_n^2] can be approximated by the formula
(a_3 /5)(a_4 /7)(a_5 /11)…(a_n /P_n)(X) for 3 ≤ m ≤ n, a_m = P_m –2 .
We establish a lower bound for TPAn (3/5)(5/7)(7/9)…(P_n–2)/P_n)(X) = 3X/P_n < TPAn.
We exhibit a formula showing as P_n increases, the number of twin primes in the interval [P_n, P_n^2] also increases. Let P_n – P_(n-1) = c. For all n (TPAn-1)(1+(2c –2)/2P_(n-1)+(c^2–2c)/2P_(n-1)^2) < TPAn

**Comments:** 2 pages of exposition and 7 pages with supporting tables. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

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### Submission history

[v1] 2019-02-01 10:53:12

[v2] 2019-02-13 08:29:18

[v3] 2019-02-16 12:01:31

[v4] 2019-02-19 08:32:39

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