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.
(This question has been killed in the comments, but it is still lacking the useful pointers.)
This is a weak form of the Hardy-Littlewood Conjecture which moreover predicts an asymptotic density for the number of such prime k-tuplets. Special cases of this include: twin primes, cousin primes, sexy primes, prime quadruplets, quintuplets, and sextuplets. While your conjecture is much weaker than Hardy-Littlewood, Kevin Buzzard's trick in the comments shows that it globally implies the infinitude of prime k-tuplets for any admissible pattern.
As far as I know, the infinitude of prime k-tuplets is an open problem for all fixed admissible patterns with k ≥ 2. Note that the Green-Tao Theorem falls short of proving any instance of this since the step size of the arithmetic progressions is not fixed. (Even the Tao-Ziegler Theorem falls short since the polynomials are required to have vanishing constant term.)