TITLE: Intervals that are free of primes
QUESTION [21 upvotes]: How can I prove that exists intervals as large as I want that are free of primes?
 
I mean, $\forall \ k \in \mathbb{N}, \exists \ k$ consecutive positive integers none of which is a prime.

REPLY [3 votes]: I would have formulated it another way. Using the contraposit of what we want to prove, there is a k for wich there is no sequence of k consecutive non-prime number. This lead to number of prime at least $\frac{n}{k}$ ie, $O(n)$. This is in contradiction with the prime number theorem suggested by Harold.