I did a research project a while ago on Odd perfect numbers (as in they actually paid me to do original research. Turns out my research wasn't so very original basically everything I thought up has been thought up before. While they aren't actually paying me for it anymore I have been continually trying to figure out some sort of novel approach to the problem. Today I was thinking about odd perfects instead of working on my electronics homework and the following equation found its way onto my page,
ln(2) = sum(ln(Pi-(1/Pi)ai)-ln(Pi-1))
Which is actually a rather unremarkable equation since in the end it ultimately says something rather akin to the sum of the reciprocals of the divisors of the OPN is equal to 2. The reason putting this equation together so intrigues me is that it brings up a question of how difficult it can be to determine whether or not a given equation has any solutions at all. This more complicated equation made me realize that I don't even know when the equation sum(1/xi)=c has solutions. What series of reciprocals can I add together to get 5? does there exist any set of reciprocals that will give me 5? Without having even begun to think about proving it I think that any rational number can be expressed in terms of a sum of reciprocals of different integers. After all the sum of the reciprocals diverges so there is no number too large to be approximated by them and they converge to zero so there is no number that they would skip over so to speak. Of course that isn't a proof and there very well may be some sort of bizarre number that cannot be written as a sum of reciprocals for some strange reason just like there are ratios of quantities that can't be written as the ratio of two integers (the ratio of the diagonal of a square to its side being the famous one of course). Assuming for the moment that we can express any rational number as a sum of reciprocals of integers then the natural question arises is there more than one way to do it? I know that 1/2 + 1/3 = 5/6 but can I break 5/6 into a combination of the reciprocals of any other two or more numbers? Furthermore how does one go about finding out how many reciprocals is the minimum necessary to compose some number? How many reciprocals does it take to add up to 19/3? or can you at all?