What are all [ Editor: "Can you determine all" in the original ] the "perfect" numbers between 1 and 1,000,000?

1   6
2   28
3   496
4   8,128
5   33,550,336

You'd be famous if you found an odd
perfect number, [ Editor: no comma in the original ] although no-one has proved that there isn't one yet!

-- J. Moran <morano@argonet.co.uk>

Perfect numbers (at least the even ones) are related to Mersenne
primes and have the following form:

 Even perfect number = (2^(n - 1)) *  (2^n - 1)
                                      ^^^^^^^^^
                                    Mersenne prime [ Editor: "Prime"
                                    in the original ]

To date [ Editor: December 1997 ] 36 values have been found for n which
make 2^n - 1 a prime.  See the following web page for the [ Editor:
"36" omitted here ] known Mersenne primes:

http://www.utm.edu:80/research/primes/mersenne.shtml#known

The largest Mersenne prime has nearly 900,000 digits.

-- phil kenny <pkenny@ix.netcom.com>

How about this one: 2305843008139952128

-- RM Mentock <mentock@mindspring.com>

A perfect number is one that all its divisors, except itself, add up to
itself.  Examples follow:

                 6 = 1 + 2 + 3
                28 = 1 + 2 + 4 + 7 + 14
               496 = 1 + 2 + 4 + . . . + 248
             8,128 = . . .

These four were know by the ancients who also believed, considering there
was one 1-digit, one 2-digit, one 3-digit, and one 4-digit perfect
numbers, that the next perfect number would have 5 digits.  But the next
three are

        33,550,336    8 digits
     8,589,869,056   10 digits
   137,438,691,328   12 digits

And the number of digits in the next few really start increasing:

   19, 37, 54, 65, 77, 314, 366, 770, 1327

The 30th perfect number is said to have 130,100 digits.

The information on odd perfect numbers that I have is from the Dr. Crypton
column in Science Digest, December 1985.  So it's a bit dated.  It queries:
Are there any perfect numbers not generated by Euclid's formula?
(Euclid's formula is  2^(n-1) * (2^n - 1) )  Euler proved that this
formula will yield all the even perfect numbers, but so far as I've heard,
no one has shown whether there are perfect numbers not generated by this
formula.

An odd perfect number, if one exists, must be evenly divisible by at least
eight different prime numbers, the article explains, of which the largest
must exceed 300,000 and the second largest must exceed 1,000.  If it is
not divisible by 3, it must be divisible by at least 11 different prime
numbers.  It also stated that an odd perfect number must leave a remainder
of 1 when divided by 12 and a remainder of 9 when divided by 36.

It goes on to say that the more constraints there are on an odd perfect
number, the less likely it is that one exists.  In 1973, Peter Hagis Jr.
proved that there is no odd perfect number below 10^50.  Since that time,
number theorists have pushed the bound below which an odd perfect number
cannot exist to above 10^200, though these latter proofs are suspect.

-- donny cherf <cherf@serg.sdsu.edu>, June 1992

Well, if that's true, then there are no odd perfect numbers, since any number
that's leaves a remainder of 9 when divided by 36 will also leave a remainder
of 9 when divided by 12.  Care to try again?

-- Carl J Lydick <carl@SOL1.GPS.CALTECH.EDU>

Sorry, I was just typing in what I read from the column, obviously without
thinking.  Looking back at the article, though, that's exactly what it
says, so perhaps they meant to say some other combination of numbers and
that's an error (or perhaps they were saying there are none).

-- donny cherf

The article should in fact have said "or", not "and".

-- Adam M. Logan <amlogan@phoenix.Princeton.EDU>

Richard Brent of the Australian National University [Editor: "University
of Australia" in the original] is doing work (or has done work) on this
problem.  He upped the lower bound for an odd perfect number to
somthing like 10^300.  So far it appears unlikely that a proof about
noexistance is forthcoming.

-- Dik T. Winter <dik@cwi.nl>