In the universe, there exist so-called “black holes”; as long as you step within their event horizon, you can never get out (unless you can exceed the speed of light). In mathematics, there are similar rules: as long as a natural number is substituted into such a rule, it will without exception fall into an infinite loop; these are known as “number black holes.” There is one “number black hole” that is particularly fascinating, so much so that some have called it a “conspiracy attempting to slow down the progress of mathematics in the United States”—this is the “Hailstone Conjecture.”
The Hailstone Conjecture:
Choose any natural number. If the chosen natural number is even, divide it by 2; if it is odd, multiply it by 3 and add 1. If the transformed natural number becomes even, divide it by 2 again; if it becomes odd, multiply it by 3 and add 1. Continue this process, and eventually, they all “return to their roots”—becoming 1.
Example (an automatic verification program is provided below):
11 \to 34 \to 17 \to 52 \to 26 \to 13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1
Reference:
http://baike.baidu.com/view/736196.htm
(Note: Baidu’s information is quite messy and much of it is meaningless; just take a quick look ^_^)
Below is a verification program:
You can pick any example (I recommend you choose 27, haha), and you will see that a natural number changes according to the rules, fluctuating in size, and finally stopping at 1. This process is very similar to the formation of hailstones: starting as a water droplet, moving up and down within the clouds, fluctuating in volume, and finally turning into a hailstone that falls from the sky. Therefore, some people call the phenomenon of natural number transformation described above the “Hailstone Conjecture.”
Is the “Hailstone Conjecture” correct? Nobuo Honda of the University of Tokyo used a computer to verify that it holds true for at least all numbers below 10^{12}. However, no one has proven it yet.
This conjecture was introduced to Japan by Shizuo Kakutani, so it is known in Japan as “Kakutani’s Conjecture.” However, it is no longer clear who first proposed this conjecture. Nevertheless, we look forward to a brilliant mind who can provide a complete proof for it!
Historical Report:
One day in 1976, The Washington Post reported a piece of mathematical news on its front page. The article recounted the following story:
In the mid-70s, on the campuses of prestigious universities across the United States, people were playing a mathematical game day and night, as if possessed, neglecting sleep and food. This game was very simple: write down any natural number N and transform it according to the following rules:
If it is an odd number, the next step becomes 3N+1.
If it is an even number, the next step becomes N/2.
Not only students but even teachers, researchers, professors, and scholars joined in. Why has the charm of this game endured? Because people discovered that no matter what number N is, it eventually cannot escape returning to the bottom 1. To be precise, it cannot escape falling into the 4-2-1 cycle at the bottom; it can never escape such a fate.
When reposting, please include the address of this article: https://kexue.fm/archives/34
For more detailed information on reposting, please refer to: Scientific Space FAQ