Infinitely many composites

In number theory, we frequently ask if there are infinitely many prime numbers of a certain type. For example, if n is a natural number: (i) Are there infinitely many (Mersenne) primes of the form 2 n − 1? (ii) Are there infinitely many primes of the form n 2 + 1? These problems are often very difficult and many remain unsolved to this day, despite the efforts of many great mathematicians. However, we can sometimes comfort ourselves by asking if there are infinitely many composite numbers of a certain type. These questions are often (but not always) easier to answer. For example, echoing (i) above, we can ask if there are infinitely many composites of the form 2 p − 1 with p a prime number but (to the best of our knowledge) this remains an unsolved problem. Of course, it must be the case that there are either infinitely many primes or infinitely many composites of the form 2 p − 1 and it seems strange that we currently cannot decide on either of them.