How does Terence Tao's mind work

Collatz problem almost proven for almost all numbers

The Collatz conjecture has driven quite a few mathematicians almost insane. It is often referred to as the simplest problem in mathematics because the question is astonishingly simple. But experts warn against delving too deeply into it. Because like a siren song it pulls people under its spell. Many get lost in the problem and never find out.

But at the beginning of 2019 one of the most renowned mathematicians in the world once again dared the conjecture - and produced one of the most important results in decades: On September 8, Terence Tao published a proof that the Collatz conjecture at least for almost all numbers true is.

This article is included in Spectrum - The Week 04/2020

A simple puzzle

Although there is no written evidence, it is believed that the German mathematician Lothar Collatz posed the problem named after him in the 1930s. In fact, it sounds like a party trick: pick any number; if it's odd, multiply it by 3 and add 1; if it is even, then divide it by 2. The result is inserted again into this simple arithmetic rule and the procedure repeated over and over again.

One would probably expect that the resulting results would depend heavily on the number chosen initially. For some starting values ​​you could end up at one at some point. Other numbers, on the other hand, could produce ever larger values ​​and grow to infinity. But Collatz predicted that this would not happen. He suspected that every starting value inevitably leads to one at some point. As soon as this is achieved, the rules of the Collatz problem result in an endless, repeating loop: 1, 4, 2, 1, 4, 2, 1, ...

“It is easy to be drawn into these big, famous problems that are far beyond anyone's ability. You can waste a lot of time doing that. ”Terence Tao

The beguiling simplicity of that assumption attracted a number of amateur mathematicians and puzzle lovers over the years. “You just have to be able to multiply a number by three and divide by two and you can play around with it. It's very tempting to try your hand at it, ”explains Marc Chamberland, mathematician of Grinnell College. Researchers now have trillion (1018) tested from seed values ​​- all follow Collatz's prediction. There are many "Collatz calculators" online that can be used to check various numbers using the simple procedure. Plus, it is teeming with false evidence of amateurs claiming to have solved the problem one way or another.

Impossible proof

In the 1970s, mathematicians were able to show that almost all Collatz sequences (the list of numbers you get by following the process) eventually lead to a result that is less than the starting value. Such evidence reinforces the suspicion that Collatz was correct. In the coming decades, however, there was no significant progress. Therefore, many experts are of the opinion that the assumption is beyond the reach of today's understanding - and that it makes more sense to devote your time to other topics.