Exercise 4.2.17 ($q$ is prime if and only if $\sigma(q) = q+1$)

Proof. Let $q$ be a positive integer.

If $q$ is prime, the only divisors of $q$ are $1$ and $q$, where $q\ne 1$, thus $\sigma (q)=q+1$.

Conversely, suppose that $\sigma (q)=q+1$. Since $\sigma (1)=1$, $q\ne 1$. Assume, for the sake of contradiction, that $q$ is not prime. Then, by definition of a prime number, $q$ has a divisor $d>0$, where $d\ne 1,d\ne q$. Therefore

This contradicts $\sigma (q)=q+1$. This proves that $q$ is prime.

In conclusion, $q$ is prime if and only if $\sigma (q)=q+1$. □