Proper divisors of 1?

Question

What are the proper divisors of 1? I understand proper divisors do not include the number itself, but is 1 an exception or does it have no proper divisors?

Answer 1

It depends on the structure and/or your definition of proper divisor.

• If you say that a divisor $d$ of $b$ is proper if $d \neq b$, then the proper divisors of $1$ are exactly the invertible elements/units of the ambient structure except $1$. That is none for $\mathbb{N}$, $-1$ for $\mathbb{Z}$, $-1,i,i$ for $\mathbb{Z}[i]$ and so on.

• If you say that a divisor $d$ of $b$ is proper if $dc=b$ with a non-invertible $c$, that is $d$ is not associated to $b$, then $1$ has no proper divisors.

Personally, I would go for the second definition.

Answer 2

$1$ is a special case for almost everything. In this case, because it is the only natural number with no proper divisors, just as your definition implies. General rule with number theory: $1$ is always weird.