TITLE: Infinitude of primes in 10 consecutive integers
QUESTION [9 upvotes]: Do there exist infinitely many sets of 10 consecutive positive integers where exactly one is a prime? 
By Dirichlet's Theorem, if $a$ and $d$ are relatively prime, then there infinitely many primes in the arithmetic sequence $a+d, 2a+d, 3a+d, \cdots$.
Let $n+1, n+2, n+3, \cdots, n+10$ be ten consecutive integers, then I want to construct 9 composite integers (say $n+1$ up to $n+9$) and a prime $n+10$, but then I have no idea how to proceed.

REPLY [7 votes]: A totally different proof: 
Assume there is an M > 0 such that there is no prime p > M with p+1, p+2, ..., p+9 all composite. There are two possibilities: 
Either there is no prime > M at all. That's false because there is an infinite number of primes. 
Or there is a prime p > M, then another prime p1 among p+1, ..., p+9, then another prime p2 among p1+1, ..., p1+9 and so on. Which means lim inf (pi (n) / n) ≥ 1/9, but we know that the limit is indeed 0. 
This also works for any length of the interval.