The Corndog Conjecture
I have written extensively on The Collatz Conjecture, that seemingly simple yet deceptively difficult problem conjured up by Lothar Collatz in 1937. Simply put, take a positive whole number; if it is odd, multiply it by three and add one. If it is even, then divide it by two. Collatz suggested that any number you can think of, even one a million digits long, will work its way back to one. Lots of serious mathematicians have worked on this problem; I would say that any professional mathematician has at least taken some time to think about it, and no one knows if the conjecture is true or false.
I was shocked when, about a year ago, I came across a paper by the great UCLA mathematician Terrance Tao. His paper, while not proving the conjecture, shows that the conjecture is true for almost all numbers, and if there are numbers that don’t fit the pattern, then those numbers are very rare.
Tao’s paper is remarkable. He readily admits that his approach will not lead to a proof, but it represents the best work done on the conjecture. I think few would argue that Tao’s work is the greatest advancement ever seen on the problem.
A year or so ago, I had a couple of my nephews over. We went upstairs to the computer/math/science lab I built for them. While we were there, I asked Corndog to play around with the variables in the Collatz Conjecture. Instead of 3n+1, what would happen if it were 5n+1, or even 7n+1? Well, he opened up Scratch and changed 3 to 4. He hit the green flag and let the program run. Here are the results:
5
21
85
341
1365
5461
21845
87381
349525
1398101
5592405
22369621
89478485
357913941
1431655765
5726623061
22906492245
91625968981
366503875925
1466015503701
5864062014805
23456248059221
93824992236885
375299968947541
1501199875790165
6004799503160661
24019198012642644
12009599006321322
6004799503160661
24019198012642644
12009599006321322
6004799503160661
24019198012642644
12009599006321322
6004799503160661
24019198012642644
12009599006321322
6004799503160661
24019198012642644
12009599006321322
As you might have noticed, the output cycles between three truly large numbers. Interesting…that is a pretty good find for a first run.
As the day went on, Z kept working in an arcade we are building, and Corndog experimented with different numbers in his Collatz code. I must let everyone know that I put a special lab notebook on one of the desks. In that book, we were to put any and all notes about the experiments we were running or the robots we were building with our Raspberry Pi computers. I wish Corndog had used it. Why? He put in a number, let the program run, and then watched some ninja videos on his phone. After a substantial period of time, he glanced over at the monitor and said, “Hey, it is back where it started!” And it was.
The iterations had produced an outrageously large number and then returned to the starting point. Which number was that? I have no idea. Corndog didn’t write it down, and I was sure I would not forget it when he told me what it was. You guessed it, I forgot, and Corndog has no idea what the mystery number is.
You would think that wouldn’t be much of a problem, right? Simple experimentation should get us back to that number. It should, but it hasn’t. I spent some time trying to find it, and I can’t. It appears to have disappeared into the ether along with Corndog’s mysterious code.
I decided to write about this fiasco because it reminded me of a story from my graduate school days. Way back then, and I mean way back, we all used notebooks to scribble in when we sat in class. There were no decks of beautiful slides. There were no whiteboards; if the professor didn’t have chalk, we were all out of luck.
All those decades ago, I was famous for writing everything the professor said down in my book. Even if it was a tangential story, I would include it because I found that the story would help me remember the stuff that would appear on the test. It all formed one big link for me. When I had lots of stuff to memorize, and trust me, I often had lots of stuff to memorize, this method worked well.
As the story goes, one day, I was sitting in a Ph.D. level evolutionary biology class. The professor, a very bright man indeed, started talking about…well, I can’t remember. And when he said, “And this would make for an important paper. It wouldn’t be hard to write, either. All the research has been done; it just needs to be synthesized.” I knew he was right. This is perfect for my buddy Scott and me. We could probably have a rough draft together in a weekend. Note, I said, “I thought,” and not, “I wrote.”
You guessed it, Scott and I talked about the paper a few days later, but neither of us could remember the topic. We went to the professor, and he rubbed his chin and said, “I can’t remember; it is gone.” And it was gone. We had all forgotten the subject of this paper that would have gotten us all fantastic prizes, wonderful gifts, and dates with European supermodels. I checked my notebook, and there wasn’t a mention of it. And so it goes…
It is fairly apparent that I write a lot. I am always getting ideas; back in the good old days when I was able to run, I would think out entire chapters during a single run. Almost all of The Athena Chapters were written this way. Now that I can’t run, the ideas trickle in whenever they want. They are not anxious to be discovered; they tend toward the shy side.
That said, I still try to keep a pen and paper with me at all times. I also have been known to email myself with whatever brilliant idea pops up when I am in line to get a hamburger.
As unbelievable as it may sound, I just got an idea for a chapter in a novel I am writing. I have a yellow pad beside me; in the middle of the page, I just wrote: “Sasquatch McGuine meets the cops at the cafe. He tells them he is working on an app, a special app, a Sasquatch detecting app.” I didn’t write this next part because I believe it is understood: “hilarity ensues…” Not only that, I just figured out Sasquatch has a wife named Gertrude, I think she might be a member of Congress, or maybe she just lost her reelection bid. Or maybe, just maybe, she falls down a Collatz Conjecture bunny hole, never to be seen again.
