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Perfect-Numbers

Determines if a positive integer is a perfect number

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, i.e. the sum of its positive divisors excluding the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors.

It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist.

The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. This is followed by the perfect numbers 496 and 8128.

These define three classifications of positive integers based on their proper divisors. Let P(n) be the sum of the proper divisors of n, where the proper divisors of n are all positive divisors of n other than n itself.

• if P(n) < n then n is classed as deficient.
• if P(n) == n then n is classed as perfect.
• if P(n) > n then n is classed as abundant.