Continuing with this week's theme of simple yet unproven mathematical problems, today I thought I would discuss perfect numbers. Although they appear to be simply counting numbers, they have some very unique properties and mysteries.

First though I must apologize for part of yesterday's article. As I wrote in the introduction, my aim was to provide three simple problems in number theory. And while readers did enjoy both the Erdos-Strauss conjecture and Pillai's Conjecture, it would seem that many of you found Grimm's Conjecture to be a little too complicated and confusing for the tone of the article. It is one that I quite enjoy studying, but in hindsight it probably is more technical than most people would have liked. I apologize for that, and I will try to post a few more less complex number theory problems in the future to make up for it.

Now back to perfect numbers.

A number is said to be perfect if it is equal to the sum of its proper divisors, not including itself. For example the number 6 can be divided by 1, 2, and 3, and as it happens 6 = 1 + 2 + 3. Similarly, the number 28 is divisible by 1,2,4,7, and 14, 28 = 1 + 2 + 4 + 7 + 14. So both 6 and 28 are referred to as perfect numbers. However there are also several interesting and yet unsolved problems related to perfect numbers.

Are They Always Even? With modern computers we can easily find thousands of perfect numbers in just a few minutes. And yet every single one that has ever been discovered has been even. So where are all of the odd perfect numbers? No one has ever found one, and yet no one has ever been able to prove they do not exist.

How Many Are There? This is another intriguing question in number theory. There are arguments suggesting that there must be a largest perfect number, and beyond that there are just too many divisors to add up to the original number. There are also arguments that say there should be no limit to the number of perfect numbers. So as of this moment it is an open question in mathematics as to whether there is a largest perfect number or whether they just go on to infinity.

Is There Such A Thing As A Quasiperfect Number?  A quasiperfect number is one that is nearly perfect, but not quite. A number is called quasiperfect if the sum of its divisors, not including itself or the number one, sums to the original number. (A perfect number includes 1 in the sum, a quasiperfect number does not). There are several examples of numbers whose divisors (excluding one and itself) sum to one less or one more than the original number, but there is not a single known example in which it sums to exactly the original number. There is no reason quasiperfect numbers should not exist, and yet no one has ever found one.

Are There Any Odd, Weird Numbers? And finally, moving away from perfection and into weirdness, there is a fourth related problem on the existence of odd weird numbers. A number is called weird if its divisors (not including itself) sum to something greater than the original number, and yet have the property that no subset of its divisors sums to the original number. As with perfect numbers, several weird numbers are known but they are all even. Do odd weird numbers exist, and if not then why do they not exist?

So there you have it. A concept as simple as a perfect number still leads to all of this uncertainty. And perhaps one of you will be the one to ultimately solve one or all of these problems.