Suppose that n is an integer > 1 such that:
The prime factorization of n is known
It is known that (n + 1) is a prime
Then: What can be concluded?
Among the possibilities are the following:
We can predict, at least with some non-trivial probability, the distance to the next prime after (n + 1).
We can bound away from 0 the ratio (w – t)/w, where w is the number of primes less than n and t is the number of those primes not appearing in the prime factorization of n (the idea being that if there are relatively many of them, then at least one of them would wind up being a divisor of (n + 1)).
Possibility #2 above suggests the following definition: A prime p is said to be dormant fif there exist infinitely many positive integers n such that p < n, p does not divide n, and (n + 1) is prime. Do there exist any dormant primes? This specific question sounds more interesting than the general question stated above, and so I have given this question as the title of this post.