Perfect Numbers

Date: 11/10/97 at 12:57:26
From: Juan Maria Moreno
Subject: Perfect numbers

I have found that the sums of the digits of perfect numbers greater 
than 6 always equal 1.

        28                2+8 = 10 = 1
       496              4+9+6 = 19 = 10 = 1
33,550,336    3+3+5+5+0+3+3+6 = 28 = 10 = 1

Do the sums of all the digits of perfect numbers equal 1?

33,550,336 returns 28. Will the next perfect number after 33,550,336 
return 496?

Thanks.

Date: 11/10/97 at 14:02:43
From: Doctor Rob
Subject: Re: Perfect numbers

Yes, when you add the digits of any perfect number bigger than 6, then 
add the digits of that number, and so on, you eventually get 1. This 
is because when you divide such a number by 9, it leaves a remainder 
of 1. This is because prime numbers larger than 2 are always odd, and 
all even perfect numbers have the form n = 2^(p-1)*(2^p - 1) for a 
prime p. Since p > 2 is odd, p-1 is even, and so k = (p-1)/2 is an 
integer. Then the first factor has the form 3*a + 1, because it has 
the form

   2^(p-1) = 4^k
           = (1 + 3)^k,
           = 1 + 3*C(k,1) + 3^2*C(k,2) + 3^3*C(k,3) + ... + 3^k*
C(k,k),
           = 1 + 3*a.

The second factor is

   2^p - 1 = 2*(4^k) - 1,
           = 2 + 6*a - 1,
           = 1 + 3*(2*a).

Multiply them together, and you get

   n = (1+3*a)*(1+6*a),
     = 1 + 9*a + 18*a^2,
     = 1 + 9*(a+2*a^2),
     = 1 + 9*b.

The sum of the digits of n differs from n by a multiple of 9 because

   (10^e)*d = (10^e-1)*d + d = 999...999*d + d = 9*(111...111*d) + d.

This disposes of all even perfect numbers > 6. There are no known odd
perfect numbers, and it is conjectured that none exists, although 
nobody has yet proven that. If one exists, it has more than 100 digits 
and has a very complicated prime power factorization.

You omitted 8128 = 2^6*127 from your list.  The digital sum is 19, 
whose sum is 10, whose sum is 1. The next perfect number bigger than 
33,550,336 is 8,589,869,056, whose sum is 64, whose sum is 10, whose 
sum is 1. After that is 137438691328, whose sum is 55, whose sum is 
10, whose sum is 1. 

I do not know if there is a perfect number whose digital sum is 496, 
but it would have to have at least 56, and probably about 110 decimal 
digits.

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