It is known that the twin prime conjecture is a special case of the $k$-tuple conjecture. See if you want the article with title k-Tuple Conjecture from the encyclopedia Wolfram MathWorld.
On the other hand I know* the Theorem 2 of , due to Suzuki, concerning a formulation of the twin prime problem.
Question. Is it possible/feasible to get an alternative formulation, in the spirit of Suzuki's Theorem 2, for different prime constellations/k-tuples? Many thanks.
If the statement is in the literature refer the bibliography, and I try to read it from the literature.
 Maria Suzuki, Alternative Formulations of the Twin Prime Problem, The American Mathematical Monthly, Vol. 107, No. 1 (January, 2000).