Prime Between $n$ and $n!$

Number Theory Level 1

Between $3$ and $3!$ , there is $5$ , which a prime.

Between $4$ and $4!$ , there is $5$ , which a prime.

Between $5$ and $5!$ , there is $7$ , which is a prime.

True or False?

For every positive integer $n \geq 3$ , there is always at least one prime $p$ with $n<p<n!$ .

Prime Between nnn and n!n!n!