Warning: There will be some “Mathyness” to this post, but I will refrain from including anything concrete and stick with vague references or analogies, so that no math knowledge is required.
When I came home from Houston a two weeks ago, I had spent the previous 4 weeks number crunching to attack a question regarding Odd Perfect Numbers. Of course, by then the data I had collected had become quite unwieldly, but that’s the nature of such beasts. As ideas are prone to do, just prior to leaving Texas, I had had an epiphany on how to improve the number crunching process. Since my dreams of a self-driving car are still a few years away, I am unable to drive and program at the same time and I had to wait several days before I got home to implement my thoughts.
Prior to this new idea, I knew it was going to take months of work to perform all the necessary calculations. I also knew a fair portion would have to be matched up by hand, but I was expecting to be able to get the computer to do a lot of that eventually. (Though that may be some wishful thinking. As I said, the data as collected was very unwieldly.) My new process had two major improvements, most importantly, it would allow the computer to always work with the smallest numbers available at any given time. A huge benefit considering some of the numbers can be over a hundred digits long. The process would also lend itself to a simple automation process for the more difficult pattern matching part of the problem that I had been doing by hand.
So I set about writing the code for my new idea and finished it in about a day. I could have tried to continue off the work I had already done, but I wanted to see how much of an improvement the new process was in comparison. Much to my delight, I had (in some sense) recovered the previous month’s worth of work in about 4 days. This past week-and-a-half has been spent tweaking the process and grinding out more data.
In truth, I have been working on this problem for over 5 months. I started with much of the current work on Odd Perfect Numbers. By that, I mean I looked at a few published papers. Then I found the papers that those papers referenced and read through those, and so on.
As I tried to explain to my students, it takes time to ingest mathematical ideas. When you watch a teacher solve a problem on the board, it’s like magic. Yet it takes a lot of work, a lot of problem solving, some experience, and time for a person solve problems in that “Magically efficient” way. Basically, I had to teach myself the necessary mathematical ideas. And in the same way it takes weeks and months for a student to learn calculus, it took me time to learn what I needed to know about Odd Perfect Numbers. So hopefully, it makes sense that I would throw out a month’s worth of work and start over.
It is important to note that this problem is about proving the non-existence of something. I doubt that any mathematician in the world thinks that an Odd Perfect Number exists, the problem is proving it. Though this question has been around for centuries, the best mathematicians have been able to do is subtly “corral” a hypothetical Odd Perfect Number; penning it in with more and more conditions in hopes that, one day, someone will be able to show that no such number can satisfy all the conditions and thus, no Odd Perfect Numbers exist.
This is my goal. I hope to add another subtle condition to the list.
At the moment, it is known that a certain internal structure of an Odd Perfect Number must be less than a certain upper bound. In particular, the sum of the reciprocal of all the primes that divide an Odd Perfect Number must be less than ln(2) = 0.693147. If one knows a little more about the Odd Perfect Number in question, then this upper bound can be lowered. For the case that I am working on, I was able to apply some results from other papers to lower my bound to .664449, which may not seem like much, but it’s quite a bit, I promise.
After two weeks of number crunching I have assembled a collection of primes that would divide an Odd Perfect Number of the sort I am looking at and their collective “weight”, for the lack of a better term, comes to 0.654786. Which means I am .009663 shy of the upper bound I am looking to break.
I’d love to say I am on the verge of surpassing the bound, and it certainly appears to be the case, but looks can be deceiving. Currently, I am adding about .0005 to .0007 every day and even these values are slowly getting smaller and smaller. I have another “small” trick or two up my sleeve, but each day will be a grind to add more and more weight. I am figuring that the process will take 2-3 more weeks, but I am resolved to see this through to the end.
Despite the growing pressure to look for a job…
