Introduction to Perfect Numbers

The history of the theory of numbers abounds with famous conjectures and open questions.Some of these have been answered some have not.This set focuses on some intriguing discussions about perfect numbers.

\(\textbf{Definition:}\)

A positive integer \(n\) is called perfect if \(n\) is equal to the sum of all of it's positive divisor's excluding itself.

\(\textbf{Example:}\) 6 is a perfect number since it's divisors \(3,2,1\) add up to \( 6\),that is, \[3+2+1 = 6\].In other words \[\sigma(6) = 1+2+3+6 = 12 = 2*6\]

In number-theoretic notation,\(\sigma(n)\) is used to denote the sum of positive divisors of a number ,hence in case of perfect numbers \[\sigma(n) = 2n\]

The first instances of the usage of perfect numbers dates back to Pythagoras and his school who tended to associate mystical properties with these perfect numbers. For many centuries Philosophers were more concerned about the mystical and religious significance of perfect numbers rather than their mathematical properties.Saint Augustine explains that although God could have created the world in an instant he preferred 6 days to symbolize perfection. The significance of this statement is left for the reader to decide.

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Introduction to Perfect Numbers

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