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### Perfect Number

```
Date: 7/21/96 at 14:57:47
Subject: Perfect Number

How can I find a perfect number? What are some reference books for
number theory?

Thank you!
```

```
Date: 8/2/96 at 13:37:21
From: Doctor Tom
Subject: Re: Perfect Number

There are hundreds of books on number theory.  Take a look in any
library or bookstore, and find one that looks interesting to you.

A perfect number is a number such that if you add up all the divisors
including 1 and the number itself, you get twice the number:

6 is perfect, since 1, 2, 3, and 6 are the divisors, and
1+2+3+6 = 12 = 2*6

28 is perfect, since 1, 2, 4, 7, 14, and 28 are the divisors, and
1+2+4+7+14+28 = 56 = 2*28.

8 is not perfect, since 1, 2, 4, and 8 are the divisors, but
1+2+4+8= 15, which is not 2*8.

If 2^n-1 is prime (has no divisors), then
(2^(n-1))(2^n - 1) is perfect. (The "^" represents an exponent.)

As examples, 2^3-1 = 7, a prime, so (2^2)*(2^3-1) = 4*7 = 28 is
perfect.

All even perfect numbers have the form above.  Nobody knows if there
are any odd perfect numbers.  If you can find one, or prove that
there aren't any, you'll be famous!

-Doctor Tom,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory
Middle School Number Sense/About Numbers

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