TITLE: Prime number between $n$ and $n!+1$
QUESTION [8 upvotes]: I am trying to prove that ($\forall \ n\in\mathbb{N}$) there exists a prime number $q$ such that $n < q \le 1 + n!$
I have made a graph with $n=0$ through $n=10$ and found solutions to all of them looking for a pattern and I see that $n!$ gets enormous fast and it becomes quite obvious that there is a prime number in between them.  
I have considered trying to prove by contradiction that $q$ does not exist on that interval, but I don't know where to go from that statement.  Could anybody help me figure it out?  I have been staring at it for hours and I can't figure out where to go.
Thank you.
http://mathforum.org/library/drmath/view/62825.html

REPLY [4 votes]: $n!+1$ isn't divisible by the integers from $2$ to $n$. Then all its prime factors exceed $n$, and there is at least one.