# 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 $4$?

### 2 responses to “When is 4 exceptional?”

1. Glen

Aren’t the special properties of the first few integers simply because they are the first few integers rather than anything intrinsic to the integers themselves? I hope that makes sense. I’m thinking that the “rules” that are true for the higher integers are somewhat predicated on an accumulation of integers “below” them.

Like

2. Good artilce

Like