There are arbitrarily large gaps in the [[Sequence]] of [[Prime Number|Primes]], given any positive [[Integer]] $n$, there are $n$ consecutive [[Composite Number|Composite]] [[Integer|Integers]]. ### Proof Consider the integers $ (n+1)! + 2 , (n+1)!+3, \cdots,(n+1)!+n+1 $ Every one of the items in this [[Sequence]] is [[Composite Number|Composite]]. $2<j\le n+1 \rightarrow j\mid (n+1)!+j$.