Are there infinitely many primes with property X

There are infinitely many conjectures asking if such types of primes also form infinite families.

Usually the answer is yes, but no one was able to prove it yet.

Collaborative mega list: math.stackexchange.com/questions/4523193/are-there-infinitely-many-primes-of-the-form-x-we-probably-dont-know

Good section also at: en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Prime_numbers