# What are perfect numbers

#### Greg Bernhardt

Definition/Summary

A perfect number is a number which is the sum of its proper divisors (half the sum of its total divisors). Even perfect numbers are a Mersenne prime times a power of two; odd perfect numbers are not known to exist.

Equations

Sum-of-divisors function:
$$\sigma(n)=\sum_{k|n}k$$
$$\sigma(p^aq^b)=\sigma(p^a)\sigma(q^b)\;\;(p,q\text{ relatively prime})$$
$$\sigma(p^a)=\frac{p^{a+1}-1}{p-1}$$

Definition of N perfect:
$$2N=\sigma(N)$$

Form of an even perfect number:
$$N=M_p(M_p+1)/2=2^{p-1}(2^p-1)$$
where M_p is a Mersenne prime.

Extended explanation

The first two perfect numbers are:

6 = 1 + 2 + 3 = $2^{2-1} (2^2-1)$

28 = 1 + 2 + 4 + 7 + 14 = $2^{3-1} (2^3-1)$

The next two are:

496 = $2^{5-1} (2^5-1)$

8128 = $2^{7-1} (2^7-1)$

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"What are perfect numbers"

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