Perfect Numbers

Date: 12/07/96 at 23:51:42
From: Ken
Subject: Perfect Number

What are the first 10 perfect numbers?  Is there a formula for getting 
a perfect number?

Date: 01/26/97 at 16:26:16
From: Doctor Reno
Subject: Re: Perfect Number

Hi, Ken! 

Perfect numbers are something that I recently became interested in. 
They have a fascinating history in mathematics. My response will be 
somewhat detailed due to the richness of the material. With this 
information, you will be able to figure out how to find all of the 
perfect numbers that you want to find!

By definition, a perfect number is a number which equals the sum of 
its factors.  In other words, 6 is perfect because the factors of 6 
are 1, 2 and 3 and 1 + 2 + 3 = 6.  28 is the next perfect number 
because its factors are 1 + 2 + 4 + 7 + 14, whose sum is 28.

Over two thousand years ago, Euclid wrote about constructing perfect 
numbers in this way: 

  "If as many numbers as we please beginning from an unit be set 
   out continuously in double proportion, until the sum of all 
   becomes prime, and if the sum multiplied into the last make 
   some number, the product will be perfect."  

This sounds complicated, but if we take it step by step, we can 
understand what Euclid was talking about!  I will take phrases from 
this long sentence and we will explore what is happening.  (In case 
the ancient lingo is a little confusing, a "unit" is 1, and "double 
proportion" means doubling, or multiplying by 2.) 

"an unit be set out continuously in double proportion":
1,2,4,8,16,32....     (1 x 2 = 2, 2 x 2 = 4, 4 x 2 = 8, etc)
...and Euclid says we can do this as many times as we please!

"if the sum":
easy! we can do this one!
1 + 2 + 4 + 8 = 16

"until the sum of all becomes prime": simply means that we add until 
we get to a prime number!

1 + 2 = 3     That's easy, isn't it!

"the sum multiplied into the last make some number"
Easier than it sounds: simply multiply the sum to the last number:
3 x 2 = 6

"the product will be perfect"
There you have it! 6 is the first perfect number!

Using this method, we can easily find the next perfect number:
1 + 2 + 4 = 7 
7 x 4 = 28: the second perfect number!

And the next......
1 + 2 + 4 + 8 = 15....oops! 15 is NOT prime, so we have to go on.
1 + 2 + 4 + 8 + 16 = 31  OK! 31 is prime, so we can use Euclid's 
method again.
31 x 16 = 496  The third perfect number.

And the next?
1 + 2 + 4 + 8 + 16 + 32 = 63   63 is NOT PRIME
1 + 2 + 4 + 8 + 16 + 32 + 64 = 127   127 is prime (it would help to 
have a list of primes, wouldn't it?!)
127 x 64 = 8,128  (a calculator will help, too!)

The fifth perfect number (I've done a little of the work for you):
1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + 2048 + 4096
 = 8191
8191 x 4096 = 33,550,336

We now have the first five perfect numbers:

                   6
                  28
                 496
                8128
          33,550,336

As you can see, it will be difficult to find the first ten perfect 
numbers....they will get very, very large!  If you need a table of 
prime numbers, I have found one at:

  http://www.utm.edu/research/primes/lists/small/10000.txt   

Do you see some patterns in these numbers? Many mathematicians 
throughout history have been fascinated by the patterns that you also 
see!  They have played with perfect numbers and come up with a lot of 
different ideas about them.

The formula for finding perfect numbers is:

2^n-1(2^n - 1), where (2^n - 1) is prime.

It is not known at this time whether there are an infinite number of 
perfect numbers or not.

Here is a web site that gives some history and references for perfect 
numbers: MathLand:
 
  http://www.sciencenews.org/sn_arc97/1_25_97/mathland.htm    

I hope you have as much fun finding your perfect numbers as I had 
helping you with your questions!

-Doctor Reno,  The Math Forum