Difference between revisions of "Deficient number"

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===Introductory===
 
===Introductory===
 
====Problem 1====
 
====Problem 1====
Prove that all prime numbers are deficient.  
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Prove that all [[prime number|prime numbers]] are deficient.  
  
 
[[Deficient number/Introductory Problem 1|Solution]]
 
[[Deficient number/Introductory Problem 1|Solution]]
  
 
====Problem 2====
 
====Problem 2====
Prove that all powers of prime numbers are deficient.  
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Prove that all [[perfect power|powers]] of [[prime number|prime numbers]] are deficient.  
  
 
[[Deficient number/Introductory Problem 2|Solution]]
 
[[Deficient number/Introductory Problem 2|Solution]]

Latest revision as of 12:39, 9 February 2018

A Deficient number is a number $n$ for which the sum of $n$'s proper factors is less than $n$. For example, 22 is deficient because its proper factors sum to 14 < 22. The smallest deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, and 17.

Problems

Introductory

Problem 1

Prove that all prime numbers are deficient.

Solution

Problem 2

Prove that all powers of prime numbers are deficient.

Solution

See Also

Perfect number

Abundant number