The broader generalization you're headed towards is https://en.wikipedia.org/wiki/Dickson%27s_conjecture https://en.wikipedia.org/wiki/Dickson%27s_conjecture (o

The broader generalization you're headed towards is https://en.wikipedia.org/wiki/Dickson%27s_conjecture https://en.wikipedia.org/wiki/Dickson%27s_conjecture (or the even more general https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H) which basically say that the twin prime conjecture is true for every linear/polynomial generalization of twin primes.