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For a prime t between 3 and 487, the period of 1/(tn) equals tn-1 times that for 1/t.
I've just spent two weeks exploring the implications of that one sentence from Dead Reckoning. Think about it for a minute yourself, if you want … or read on and find out the shocking truth in Repeat Length Again!
Looking back at the other batches of essays that have been mostly about numbers, I see I've said pretty much the same thing every time.
Enthusiasm? Check. An essay that just grows and grows? Check. A rich lode? Check. Very pleased with the result? Check. This time, though, there were two additional factors in play.
There have been a few times in the past when I've found a rich lode of software … when I've set out to write a small program, but found that every new feature I added seemed to suggest two more that were even cooler. Just for fun, here's a short list … the names will be meaningless to most everyone, but the locations of the rich lodes might be useful to others who like programming.
The lode that the original sentence led to was for me a lode both of mathematics and of software. As I understood more of the mathematics, I was able to write better programs; and as I amplified my capabilities with better programs, I was able to understand more of the mathematics. I've written programs in connection with essays before, but I've never run into the same kind of feedback loop.
The other additional factor was the internet. I like to imagine that the essays would have turned out just as well without it, but even if that's true, the internet certainly helped motivate me … it made the possibilities of the lode much more apparent, and kept me excited by how close I was to the cutting edge of knowledge. At the moment I feel like I know as much about repeat lengths as anyone in the world!
Anyway, the result of all that interaction between numbers and mathematics and software and the internet was an extremely intense experience, unlike anything I can remember except the creation of my four-dimensional maze game. I'm glad it was possible, but at the same time I'm kind of glad it's over.
@ April (2006)