Are There Infinitely Many Perfect Numbers?

Date: 12/22/97 at 11:48:50
From: Andrea Jones
Subject: Infinity of perfect numbers.

I am doing an IB extended essay in Mathematics, and one of the 
questions I'm trying to find an answer to is whether there are 
infinitely many perfect numbers. I have read various books and been 
all around the Internet, but nobody seems to have been able to prove 
anything.

There are so many theories, but nobody seems to have come up with 
anything definite.  

Thank you.

Date: 12/22/97 at 13:05:13
From: Doctor Wilkinson
Subject: Re: Infinity of perfect numbers.

Your suspicion is correct:  nobody knows whether or not there are 
infinitely many perfect numbers. As you have probably found out, all 
the even perfect numbers are of the form 2^(p-1) * (2^p - 1), where 
both p and 2^p - 1 are prime. Primes of the form 2^p - 1 are called 
Mersenne primes, and the largest known primes are Mersenne primes, but 
nobody knows whether there are an infinite number of them or not.

It is also unknown whether or not there are any odd perfect numbers.

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

Date: 12/22/97 at 13:06:03
From: Doctor Rob
Subject: Re: Infinity of perfect numbers.

Hi!  Thanks for writing to Dr. Math.

The reason you cannot find any proofs of this statement is because 
nobody knows whether it is true or false. It is one of the outstanding 
research problems of Number Theory.

You are undoubtedly aware that there are no known odd perfect numbers, 
and that it is conjectured (but not proven, yet!) that none can exist.  
What has been proven is that if there are any, they have more than 
200 decimal digits, and they have very many small prime divisors.

You are undoubtedly also aware that if p is a prime number, and if 
2^p - 1 is also a prime number, then 2^(p-1)*(2^p - 1) is an even 
perfect number, and all even perfect numbers have this form.  This 
means that the search for perfect numbers is related to the search for 
primes of the form 2^p - 1, called "Mersenne primes."  Every Mersenne 
prime yields an even perfect number, and vice versa.

To prove that there are infinitely many even perfect numbers is 
equivalent to proving that there are infinitely many Mersenne primes.  
This seems to be very hard. Most number theory experts think that 
there are infinitely many Mersenne primes. There are only 36 known, 
however, and the search for them which is underway to produce more is 
using huge amounts of computer time. See 

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

for more info on Mersenne primes and the search for them.

While it is clear that statistics has very little to do with Mersenne
numbers, the following paper applies statistics to this problem.  
Included are appropriate warnings about the assumptions made, whose 
validity is unproved, but which seem quite reasonable.

Donald B. Gillies, "Three New Mersenne Primes and a Statistical 
Theory," _Mathematics of Computation_, vol. 18 (1964), pp. 93-95.

If Gillies's assumptions were correct, there would be infinitely many 
even perfect numbers, and we would understand about how many there are 
likely to be in any given range of exponents p.  In my opinion, this 
is the closest anyone has come to proving that there are infinitely 
many perfect numbers.

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