Primality of n factorial minus one

Statement

For $n \in \mathbb{Z}^+$ set $c_n = n! - 1$ (where $n! = 1 \cdot 2 \cdots n$). Decide whether $c_n$ is prime for $n = 3, 4, 5, 6, 7$.

Solution

Compute the factorials and subtract $1$:

• $n=3$: $c_3 = 6 - 1 = 5$ → prime.
• $n=4$: $c_4 = 24 - 1 = 23$ → prime.
• $n=5$: $c_5 = 120 - 1 = 119 = 7 \cdot 17$ → not prime.
• $n=6$: $c_6 = 720 - 1 = 719$; testing primes up to $\sqrt{719}\approx 26.8$ none divides it → prime.
• $n=7$: $c_7 = 5040 - 1 = 5039$; testing primes up to $\sqrt{5039}\approx 71$ none divides it → prime.

So $c_n$ is prime for $n = 3, 4, 6, 7$ and composite for $n = 5$.