nt.number theory - Prime numbers with given difference

Let be given natural numbers N_1,N_2,N_3,...,N_k such that for every prime p less or equal
k set N_1,N_2,N_3,...,N_k does not contain all reminders modulo p. Is it right that there exists number X such that all X+N_1,X+N_2,X+N_3,...,X+N_k are prime? I think it must follow
from some theorems about prime numbers in arithmetical progression.