Divisor Sum of Prime Number

Theorem

Let $n$ be a positive integer.

Let $\map {\sigma_1} n$ be the divisor sum function of $n$.

Then $\map {\sigma_1} n = n + 1$ if and only if $n$ is prime.

Proof

From Rule of Transposition, we may replace the only if statement by its contrapositive.

Therefore, the following suffices:

Sufficient Condition

Suppose $n$ is a prime.

By definition, the only positive divisors of $n$ are $1$ and $n$ itself.

Therefore $\map {\sigma_1} n$, defined as the sum of the divisors of $n$, equals $n + 1$.

$\Box$

Necessary Condition

Suppose $n$ is not a prime.

From One Divides all Integers and Integer Divides Itself, both $1$ and $n$ are divisors of $n$.

As $n$ is composite:

$\exists r, s \in \N: r, s > 1: r s = n$

Trivially, both $r$ and $s$ are divisors of $n$.

Hence:

$\map {\sigma_1} n \ge n + 1 + r + s > n + 1$

$\blacksquare$

Also see

• Divisor Sum of Power of Prime

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1$