Q:

Prove that any power of a prime is not a perfect number

How do I prove:

Let $p$ be a prime, and $n$ be a positive integer.  Then $p^n$ is not a perfect number.

One example is when $p = 2$ and $n = 3$, the question is to show $8$ is not a perfect number.  And I know that out of $1, 2, 3, 4, 5, 6, 7, 8$, the proper divisors of $8$ are $1, 2,$ and $4$, with $1 + 2 + 4 = 7 \ne 8$, so $8$ is not a perfect number.
But how do I show this for any $n$ and $p$?

A:

The divisors of $p^n$ are $1,p,p^2,\ldots,p^n$.  The proper divisors are all but the last one.  The sum of those is
$$
1+p+p^2+p^3+\cdots+p^{n-1}.
$$
This is a geometric series.  Apply the standard formula for the sum of a finite geometric series and see if you get $p^n$.

A:

$p$ prime then, $1 +p + p^2 + \cdots + p^{n-1} \neq p^n$.

