Deficient number/Introductory Problem 2

Problem

Prove that all powers of prime numbers are deficient.

Solution

The proper factors of $p^n$ for some prime $p$ and positive integer $n$ are $1, p, p^2, \cdots ,p^{n-1}$ and their sum is $1+ p + p^2 + \ldots + p^{n-1} = \frac{p^n-1}{p-1} < p^n$

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