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

Question

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 $p$?

Answer 1

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 $p^n$.

Answer 2

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