Sunday, December 15, 2019

The Collatz Conjecture is a little closer to being proved ... say, what?

Now, I'm no mathematician (although, after producing my History of Mathematics website some years ago, I still get emails from earnest, presumably young, mathematicians asking for my professional opinions on their latest revolutionary theorem or solution...), but I do find some of this stuff interesting.
For instance, I just found out about an interesting conjecture, posited by German mathematician Lothar Collatz way back in the 1930s. The Collatz Conjecture is sometimes called "the most dangerous problem in mathematics" or "a siren song" because it appears so simple, but it has confounded the best mathematicians ever since. Many a mathematician has been tempted to solve it, only to disappear down its rabbit hole for years on end, to the exclusion of other, more profitable or more meaningful, work.
In essence, the conjecture asks you to pick any starting number and, if it is odd multiply it by 3 and add 1, and if it is even, simply divide it by 2. Rinse and repeat ad infinitum. For example: 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Or: 320, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1. The conjecture is that, given that you start with a positive whole number, the series will always run down to 1, and not spiral away to infinity. In fact, it will always end up at 4, then 2, then 1. Which is certainly what seems to happen, but the trick is proving it (which is where the serious mathematics and algebra comes in).
Anyway, the reason, this came to my attention at all is because an Australian-American mathematician called Terence Tao has managed to get a little bit closer to proving the conjecture, by proving mathematically that the conjecture is "almost" true for "almost" all numbers. While this sounds like a distinctly underwhelming and unimpressive cop-out to us non-mathematicians, apparently this is considered one of the most significant proofs on the conjecture in decades, and the kind of thing that may open the floodgates to a complete proof one day.
So, there you go, a little glimpse into the obscure world of pure mathematics.

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