Perfect Numbers - Basics, History

Date: 11/3/96 at 9:28:54
From: Anonymous
Subject: math question

What is the next perfect number after 28?

Date: 11/3/96 at 21:42:37
From: Doctor Sydney
Subject: Re: math question

It has been shown that the next biggest perfect number after 28 is 
496.  How do you suppose this was shown?  Well, one way to show it is 
to go through all of the numbers between 28 and 496 and show that 
each one is not perfect.  In other words, we can prove that 496 is 
the next biggest perfect number by showing that each number between 
28 and 496 is not perfect.  This amounts to showing that the sum of 
the factors of each number between 28 and 496 is not equal to the 
number itself.  Since this is such a big task, computers have been 
programmed to factor numbers and check to see whether or not they are 
perfect.

There are other ways to show that 496 is the next biggest perfect 
number after 28, but this is the simplest one to understand.  If you 
are curious about other ways, feel free to write back!

Did you know that no one knows whether there are infinitely many 
perfect numbers or finitely many perfect numbers?  If you want to 
know more about this, write back to us!

-Doctor Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 11/04/96 at 19:44:54
From: Anonymous
Subject: Re: math question

Thank you for answering my question about the next perfect number 
after 28.  I would be interested to learn more about how you could 
find a formula that finds perfect numbers.  I am only in 6th grade so 
I cannot understand mathematical words, but that doesn't mean that I 
am a numbskull.

Sincerely,
																	
DZINE

Date: 11/06/96 at 11:31:28
From: Doctor Yiu
Subject: Re: math question

Dear Dzine,

The search for perfect numbers began in ancient times.  More than two 
thousand years ago, the Greek mathematician Euclid explained a method 
for finding perfect numbers that is based on the concept of prime 
numbers.

A prime number is a whole number which cannot be EXACTLY divided by a  
smaller whole number.  The number 1 is not regarded as a prime number. 
So, the prime numbers begin with 2, 3, 5, 7, 11, etc. and never end.

*********************************
Euclid's method also makes use of the powers of 2, which are numbers
obtained by multiplying by 2 by itself over and over again (after you 
start with 1):

1,  2,  4,  8,  16,  32,  64,  128,  etc

(Each time you DOUBLE the previous number to get the next)

It is useful and important to note that the SUM of a string of the 
powers of 2 is equal to the next power of 2 MINUS 1.

For example,

1+2 = 3 = 4 minus 1,
1+2+4 = 7 = 8 minus 1,
1+2+4+8 = 15 = 16 minus 1,
1+2+4+8+16 = 31 = 32 minus 1,
1+2+4+8+16+32 = 63 = 64 minus 1,
1+2+4+8+16+32+64 = 127 = 128 minus 1.

This goes on and on, but I will stop here and you can try to write 
down a few more lines yourself. 

***********************************
Now, Euclid's method of finding PERFECT numbers  is this: 

Beginning with the number 1, if  you ADD the powers of 2 (as above), 
and IF the SUM is a PRIME number, then you get a PERFECT number by 
multiplying this sum to the LAST power of 2.

Examples:

If you add 1+2, the sum is 3, a PRIME number.  This means 3 x 2 = 6 is 
a perfect number.  In fact, it is the smallest perfect number.

If you add 1+2+4, the sum is 7, a prime number.  This means  
7 x 4 = 28 is a perfect number.

If you add 1+2+4+8, the sum 15 is NOT a prime number, so you can't use
Euclid's method here.

If you add 1+2+4+8+16, the sum is 31, a prime number.  This means that  
31 x 16 = 496 is another perfect number.

If you add 1+2+4+8+16+32, the sum is 63, which is NOT a prime (since 
9 x 7 = 63) so you can't get a perfect number from this.

If you add 1+2+4+8+16+32+64, the sum is 127, which IS a prime.  (It 
may take quite long time to check that 127 is indeed a prime; let's 
accept that it isjj and see what it gives you).  Well, according to 
Euclid's method, your perfect number is 127 x 64 = 8128.

******************************************
Now, to find other perfect numbers, you can continue this method.  But 
let me write down the NEXT perfect number:

33550336

and tell you a few things about perfect numbers.  

(1)  Euclid's method indeed tells you how to find ALL possible EVEN 
perfect numbers.  (An even number is a whole number that you can 
"evenly" divide by 2). 

(2)  Nobody knows whether there is an odd perfect number. (An odd 
number is a whole number that produces a remainder when you divide by 
2).
 
(3)  Because it is very difficult, even for the fastest computers, to 
decide whether or not a very, very, very big number is a prime number, 
there are only 34 perfect numbers that have been found.  The biggest 
KNOWN perfect number was only found this last September.

You can get this largest KNOWN perfect number by: 

Beginning with the number 1, keep on DOUBLING it for EXACTLY one 
MILLION 257 THOUSAND and 786 times to get a FIRST number. 

Now, double this number and subtract ONE from it to get the SECOND
number.

Multiply these two numbers together and the result is a perfect number 
which has more than 750 THOUSAND digits.

It takes more than one hundred pages of very very small print to 
actually print out this number!

Mathematicians are still trying to beat this record.

You can find more about perfect numbers at the following Web site:

  http://www.utm.edu/research/primes/mersenne.shtml   

-Doctor Yiu,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/