Does an odd perfect number exist?

Take a positive whole number, like, for example, 6. Write down all of its positive divisors except itself. For example, 1 divides 6, 2 divides 6 and 3 divides 6. Add up all these divisors. Here we get 1+2+3=6. Remarkably, we got our original number back. That was not a magic trick, in fact it doesn't usually work if you don't start with the right number, but 6 was a very special number. Any positive whole number which is the sum of all of its positive divisors, not including itself, is called a perfect number. Every perfect number anybody has ever discovered is even, but there isn't any obvious reason why this should be true. Is there an odd perfect number? Nobody knows. Maybe you can find one, or maybe you can prove that every perfect number must be even.

The image shows the first 6 perfect numbers. Source: Wikipedia contributors. "List of perfect numbers". Wikipedia, The Free Encyclopedia. 15 Dec. 2009. Retrieved 21 Mar. 2010.