## Saturday, January 23, 2010

### Unsolved: Odd Perfect Numbers

A perfect number is a number that is the sum of its proper divisors (i.e., including 1 but excluding itself).

For example, the smallest perfect number is 6.

1 + 2 + 3 = 6

The next two perfect numbers are

1 + 2 + 4 + 7 + 14 = 28
1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496

See sequence A000396 in OEIS for a list of the first ten perfect numbers.

On a related note, a number that is smaller than the sum of its proper divisors is an abundant number, while a number greater than the sum of its proper divisors is called deficient. The smallest abundant number is 12, whose proper divisors sum to 16.

All of the perfect numbers that have been discovered so far are even.

The problem is to find an odd perfect number, or prove that no odd perfect numbers exist.

If there is an odd perfect number, we already know a lot about it. It must be at least 300 digits long, with at least 75 prime factors, at least 9 of them distinct. Its largest prime factor must be greater than 100,000,000. Its second largest prime factor is greater than 10,000, and its third largest is greater than 100.

John said...

So is this a homework problem? :-)

Bill the Lizard said...

John,
Yes, this is an assignment. Good luck with it. ;)

Anonymous said...

I don't know about odd perfect numbers but there are infinitely many perfectly odd numbers 1,3,5,7...
(just kiddin)

Bill the Lizard said...

Alfred Hitchcook,
Nice. I also noticed that there are infinitely many numbers that are only off by one. 2, 4, 6, 8... :)

John (the other one!) said...

I'm surprised there isn't a way to prove all Perfect Numbers are related to Mersenne Primes. I'll take a closer look too see what I can come up with :-)

Bill the Lizard said...

John(the other one!),
Well, there is a way to show that the even perfect numbers are related to Marsenne primes, we're just not sure about all of them yet. :)

My next "Unsolved" post will talk about that relationship.

Anonymous said...

I proved that odd-perfect numbers can't and don't exist; just go to www.oddperfectnumbers.com; the answer is on the very first page opened. Bill Bouris

moka math said...

hi bill i know how to find odd perfect no bu its very larg to calculate i need some help.

moka math said...

hi bill i know how to find odd perfect no bu its very larg to calculate i need some help.