Prime Between \(n\) and \(n!\)

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 \(n<p<n!\).

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