The "Conspiracy" to Slow Down American Mathematical Progress

企图减缓美国数学进展的“阴谋” 苏剑林 · 2009-07-26 · Mathematics mystery united states flash attention black hole number

There are so-called "black holes" in the universe: once you step inside their event horizon, you can never escape (unless you can travel faster than light). In mathematics, there is a similar phenomenon: substitute a natural number into a certain rule, and without exception it will fall into an endless loop. This is called a "number black hole." There is one particularly fascinating "number black hole"—so fascinating that some have even called it "a conspiracy to slow down the progress of American mathematics"—and that is the Collatz conjecture, also known as the hailstone conjecture.

The Hailstone Conjecture:

Pick 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. When the transformed natural number becomes even, divide it by 2 again; if it becomes odd, multiply it by 3 and add 1 again. Continue this process, and eventually every number "returns to its roots"—it becomes 1. more

Example (an automatic verification program is provided below):$11->34->17->52->26->13->40->20->10->5->16->8->4->2->1$

Reference: http://baike.baidu.com/view/736196.htm

(Note: the material on Baidu is rather messy, and much of it is not particularly meaningful—just skim through it for fun ^_^)

A verification program is provided below:

You can try any example you like (I recommend 27, ha ha), and you'll see that when a natural number is repeatedly transformed by this rule, it fluctuates up and down erratically, and eventually settles down to 1. This process closely resembles how hailstones form: starting as a water droplet, it rises and falls within the clouds, its size fluctuating, until it finally becomes a hailstone and falls from the sky. That's why this phenomenon of natural-number transformation is called the "hailstone conjecture."

Is the "hailstone conjecture" true? Shizuo Kanada of the University of Tokyo used a computer to verify that it holds, at least for all numbers below $10^{12}$. However, no one has yet proven it in general.

This conjecture was introduced to Japan by Shizuo Kanada, and so in Japan it is called the "Kanada conjecture." But exactly who first proposed this conjecture is no longer clear. Still, we all look forward to the day when some brilliant mind manages to give a complete proof of this conjecture!

Historical Report:

One day in 1976, The Washington Post ran a mathematics story on its front page. The article recounted a story like this:
In the mid-1970s, on campuses of prestigious American universities, people were, as if driven mad, playing a certain mathematical game day and night, forgetting to eat or sleep. The game was extremely simple: write down any natural number N, and transform it according to the following rule:
If it is odd, the next step turns it into $3N+1$.
If it is even, the next step turns it into $N/2$.
Not only students, but even teachers, researchers, professors, and scholars all joined in. Why did this game hold such enduring appeal? Because people discovered that no matter what number N was, it could never escape—it always eventually fell back to the bottom, to 1. More precisely, it could never escape falling into the 4-2-1 loop at the bottom; there was no escaping that fate.