Perfect Numbers

Date: 08/14/97 at 18:52:01
From: Insa Thiele
Subject: Perfect Numbers

What is the highest perfect number that has been found? How many 
perfect numbers are there? What are they?

I already know that 6 and 28 are perfect numbers, and I would like to 
know the other ones.

Date: 08/17/97 at 14:18:32
From: Doctor Terrel
Subject: Re: Perfect Numbers

Dear Insa,

You've asked the "perfect" question.  I like to talk about perfect 
numbers a lot.  

You're certainly right; 6 and 28 are perfect numbers.  They are the 
first two as well.  The next three are 496, 8128, and 33,550,336.  
Really!  

Perfect numbers don't show up too often, do they? They're sort of like 
perfect people; there are not many of those either, and after that 
fifth one, they get even bigger. Soon they are so big that it's not 
easy to write them out.

To the best of my knowledge, we only know 35 perfect numbers. As of 
1966, 24 perfect numbers were known. The 24th one had 12,003 digits.  
Of course this means computers are needed to do this work.

Perfect numbers can be formed every time a prime of a certain type is 
found. Just last November a new prime of this type was found. Read the 
information below; it was taken from a math newsletter that I write 
for my students.

------------------------------------------------
NEWEST  GIGANTIC  PRIME

   Last November while working at his computer a 29-year-old Frenchman 
found the newest prime currently known.  Joel Armengaud's computer 
needed a mere 88 hours of time to prove that

   2^1398269 - 1

was prime, i.e. has no factors other than itself and 1. 

   This number is so huge that it consists of 420,921 digits.  It 
begins 8147175644 and after 420,901 more digits it ends 8451315711.
 
   TMN printed out this number from the Internet; it required 85 pages 
of paper!  If we write it in scientific notation, we have

  8.147175644 x 10^420920

  Since this prime is one of a class of primes known as Mersenne 
primes [which means of the form 2^p - 1, where p is a prime], we now 
have a new "perfect number" to add to our collection.  It is

  2^1398268 x (2^1398269 - 1)

and it consists of a mere 841,842 digits! You can find out more about 
this fascinating subject on the Internet at  

  http://www.utm.edu/research/primes/index.html   

----------------------------

That last part tells us that whenever a new Mersenne prime is found, 
a new perfect number also is found.  We use the formula 

   2^(p-1) * (2^ p - 1)

to do this.

Check out that website to find out more about primes and perfect 
numbers. Have fun!

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