TITLE: Finding how many prime numbers lie in a given range
QUESTION [7 upvotes]: How many prime numbers $p$ are there which satisfy this condition?
$$13! +1 \lt p \leq 13! +13$$
Which method should I use to solve this, or could you help with the first steps?

REPLY [2 votes]: What you have to remember is that $n!$ is divisible by each prime number $p \leq n$. Then $n! + p$ is also divisible by $p$.
In the specific case of $n = 13$, it follows that $13!$ is divisible by $2, 3, 5, 7, 11, 13$, and consequently, $13! + 2$ is divisible by $2$, $13! + 3$ is divisible by $3$,  you get the idea.
So yeah, no primes there.