We prove similar theorems on twin primes for our Sequence A167493 [5] which
is "dual" to earlier considered Sequence A166944. Besides, we consider several
new sequences of such type which are also connected with twin primes. Finally,
we give a simple recursive algorithm for receiving twin primes which is
important with the point of view analysis of the structure properties of the
twin prime sequence.

We prove the theorems which are equivalent to the Roland's results such that
a new form of them allows to consider some generalizations. In particular, we
give generators of primes more than a fixed prime.

Let $p_n$ is the $n$-th prime. With help of the Cram\'{e}r-like model, we
prove that the set of intervals of the form $(2p_n,\enskip2p_{n+1})$ containing
at list 3 primes has a positive density with respect to the set of all
intervals of such form.

We give two proofs of the identity $$\sqrt{\frac{\cos\frac{2\pi}{5}}
{\cos\frac{\pi}{5}}}+\sqrt{\frac{\cos\frac{\pi}{5}} {\cos\frac{2\pi}{5}}
}=\sqrt{5},$$ using and not using the gold ratio.

Let p be an odd prime, such that p_n<p/2<p_{n+1}, where p_n is the n-th
prime. We study the following question: with what probability does there exist
a prime in the interval (p, 2p_{n+1})? After the strong definition of the
probability with help of the Ramanujan primes and the introducing
pseudo-Ramanujan primes, we show, that if such probability P exists, then
P=2-\sqrt{2}=0.585786.... As a corollary, we show that if probability P exists,
then the probability, that the interval (2p_n, 2p_{n+1}) contains a prime,
exists as well and is 2(\sqrt{2}-1)= 0.828427...