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2 fixed theorem name

(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 Polynomial Green-Tao Tao-Ziegler Theorem falls short since the polynomials are required to have vanishing constant term.)

1

(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 Polynomial Green-Tao Theorem falls short since the polynomials are required to have vanishing constant term.)