  We show that there exists a bounded pattern of m consecutive primes for any
m>0, that means a tuple H_m of m distinct non-negative integers h_i
(i=1,2,...m) such that its translations contain arbitrarily long (finite)
arithmetic progressions. More precisely, the set of natural numbers n for which
all components n+h_i (i=1,2,...m) are consecutive primes contains arbitrarily
long (finite) arithmetic progressions. Moreover, the set of m-tuples that
satisfy this property represents a positive proportion of all m-tuples. The
present result is the generalization of the results of Green-Tao (about the
existence of arbitrarily long arithmetic progressions) and of Maynard/Tao
(about the existence of infinitely many bounded blocks of m primes, where m is
an arbitrary natural number). It also generalizes the author's work which first
showed the existence of infinitely many Polignac numbers and they contain
arbitrarily long (finite) arithmetic progressions (arXiv: 1305.6289v1, 27 May
2013) which was a common generalization of the above mentioned result of
Green-Tao and that of Zhang (about the exisatence of infinitely many bounded
gaps between primes).
