  In the present work we prove a number of surprising results about gaps
between consecutive primes and arithmetic progressions in the sequence of
generalized twin primes which could not have been proven without the recent
fantastic achievement of Yitang Zhang about the existence of bounded gaps
between consecutive primes. Most of these results would have belonged to the
category of science fiction a decade ago. However, the presented results are
far from being immediate consequences of Zhang's famous theorem: they require
various new ideas, other important properties of the applied sieve function and
a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, and
Zhang, respectively.
