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. 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.


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