Gaps between prime numbers can be arbitrary large.

(to the proof gap between prime numbers arbitrary large we must show. we can find the arbitrarily large set of consecutive non-primes)

Proof:    In order to find a gap of length n between two consecutive primes, consider the following list of consecutive integers:

(n+1)! + 2,    (n+1)! + 3,    (n+1)! + 4,  …, (n+1)! + n,    (n+1)! + (n+1)

we know that the first number in that list is divisible by 2, the second number is divisible by 3, the third number is divisible by 4, … and the last number is divisible by n+1.  So all of these numbers are non‐primes! by selecting any n we can create arbitrary large consecutive non-primes