I'm currently going through "An Introduction to the Theory of Numbers" by Hardy and Wright and at one point, they discuss why the distance from one prime to the next must have a long chunk of composites in between. I'm trying to understand the reasoning but I'm having some difficulty. Here is what they say:
Suppose that $2, 3, \ldots, p$ are the primes upto $p$. Then all the numbers up to $p$ are divisible by one of these primes, and therefore, if $2\cdot3\cdot5 \cdots p = q$ all of the $p-1$ numbers i.e. $q + 2, q+3, q+4 \ldots , q+p$ are composite.
I'm having particular difficulty understanding the $q + 2, q+3, q+4, \ldots , q+p$ part. If $q$ is equal to the product of primes from $2$ to $p$ then how can how can we have all the $p-1$ numbers be equal to $q+2, q+3, q+4, \ldots q + p$?
Can someone please clarify this? or if not, just explain what they are trying to say here.
Thanks a bunch!