Some News About The Collatz Conjecture
As many of you know, I spend a fair amount of time writing about and researching The Collatz Conjecture. I have a computer running 24/7 whose only purpose is to find a counterexample to this insanely complex yet straightforward problem. For those of you unfamiliar with the conjecture, it couldn’t be more simple. Take any positive integer; if it is even, divide it by 2. If it is odd, multiply it by 3 and then add 1. In 1937, Lothar Collatz proposed that any positive integer subjected to this process would end up at 1.
No one has proven this conjecture, and young mathematicians are strongly encouraged not to work on it. Many people believe the mathematics required to solve it has not yet been discovered. It really is a curious, impenetrable problem.
I have written about the surprising progress the great mathematician Terry Tao made a couple of years ago on this topic. He proved that the conjecture is true for nearly all numbers; if there are counterexamples, they are rare. It truly is an astonishing piece of work.
This short post is about the loop that the numbers make at the end of the sequence. If a number reaches 1, what happens next? By rule, it turns into a 4, which turns into a 2, which becomes a 1. And then, of course, the infinite 4-2-1 loop continues.
I came across a video the other day on the Veritasium channel. They do outstanding work over there; we all should subscribe. Watching the video, you will learn that the collective “we” has tested every number up through 2 raised to the 68th power. They all return to 1 via the 4-2-1 loop. What came next shocked and discouraged me. Mathematicians have concluded that if a loop other than 4-2-1 does exist, it must be larger than 186,000,000,000 numbers long. Huh? Well, all right then.
After playing that section of the video multiple times just to make sure I heard correctly, I decided that I am still going to build my own Raspberry Pi “Super Computer” to attack this problem. If a novel loop is calculated, I doubt I will ever be able to spot it. There is a chance, though, that I could still discover that elusive counterexample, and that would be very cool. After all, mathematical immortality awaits the person who finds one of those vast numbers (if any really do exist).
NOTES:
I once heard someone explain what mathematical immortality really is. He said that if you are a mathematical immortal, that means that one day (probably 500 years or so from now), a student will look up your name to include it in a research paper that only their professor will read. Not bad, not bad at all…
