Difference between revisions of "Deficient number/Introductory Problem 2"
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| − | The proper factors of <math>p^n</math> for some prime <math>p</math> and positive integer <math>n</math> are <math>1, p, p^2, \cdots ,p^{n-1}</math> and their sum is <math>1+ p + p^2 + \ldots + p^{n-1} = \frac{p^n-1}{p-1} | + | The proper factors of <math>p^n</math> for some prime <math>p</math> and positive integer <math>n</math> are <math>1, p, p^2, \cdots ,p^{n-1}</math> and their sum is <math>1+ p + p^2 + \ldots + p^{n-1} = \frac{p^n-1}{p-1} < p^n</math> |
Latest revision as of 19:53, 4 March 2020
Problem
Prove that all powers of prime numbers are deficient.
Solution
The proper factors of 
for some prime 
and positive integer 
are 
and their sum is 
