# When is 4 exceptional?

What makes the number $4$ interesting?

Obviously there will be many answers to this. Here are a couple of special things about $4$ which have turned up in my first year module recently (and in one case this was unplanned!)

• $4$ is the only square number which is exactly one more than  a prime number
• $4$ is the only positive integer $n$ such that $n$ is not prime, and yet $(n-1)!$ is not divisible by $n$. (Here we follow the convention that $0!=1$.)

Do you have any other favourite exceptional properties of $4$?