# Collatz ject ha in my mind, almost

I realized it would scare you or grab your attention. Previously though, bear with me a few moments, the instructions in it are very simple, they are very short:

1) Think of any positive integer.

2) If odd, multiply 3 and add 1.

3) If it is equal, divide by 2.

4) Go back to # 2 and repeat.

So if you start with 5, you will have this sequence: 5-16-8-4-2-1-4-2-1 …

With 9, you get it: 9-28-14-7-22-11-34-17-52-26-13-40-20-10-5-16-8-4-2-1-4-2 -1 …

These scenes are called snowflake numbers because of how yo-yo they come up and down, because snowflakes fall on the clouds they form. In that previous column, I mentioned the 112-step yo-yo fest that started with 27 kicks, part 78 of which was 9,232.

You will see that the snowflakes separately, in the two cases above, reach sequence 1. At that point, we can stop because it’s just the 1-4-2-1 cycle, which brings us to Collatz. In the 1930s, the German mathematician Lothar Kollatz suggested that no matter what number you start with, the sequence should always reach 1.

It’s very simple, at least how it is pronounced. Mathematicians, however, have not been able to prove this in all these decades. They begin with numbers exceeding the 20th power of 10 to more than two hundred quintillions and each of them ends in 1, often producing a snowflake order. This is very good evidence, you think, that Kollatz’s ject ha is true. But not for mathematicians. They seek not only several quintillion examples without a single counter-example, but also ironclad, irrefutable, logical proof. Note that there is only one counter-example proof-ject that is false.

They do not have proof yet. But there has been recent and ingenious progress on one.

The story begins with a blog post by Terence Tao, winner of the Fields Medal in 2011 and is widely regarded as one of the brightest stars in contemporary mathematics. He called the post “Collatz ject ha, Littlewood-offered doctrine and the powers of 2 and 3” (bit.ly/3qWcGZm). The first few lines describe snowflake instructions and the Collatz ject ha is more mathematical than I did above. Explaining how mathematicians view it:

“Public questions with this level of notoriety are what Richard Lipton calls ‘mathematical diseases’ (and what I call an ‘unhealthy obsession with a well-known problem’).

Mathematicians have explored the Collatz canvas as one of the most fundamental of mathematical ideas: a model. There are plenty of beautiful pictures showing how the numbers behave when jumping about snowflakes-fashion. Disappointingly, there were no patterns. However, Tao spent a few days in 2011. The rest (long) post then explains his results. It has attracted over 200 comments. In August 2019, an anonymous reader left:

“Another requirement is to show that the number of integers less than N for any periodic loop is true ‘almost everywhere’ in the sense that o (N) is greater than n.”

In simple words: Probably there is progress in trying to prove ject han for all numbers, but not all. This feeling flooded Tao’s collatz juices again, because later that year, he took a few days to re-enter the ject hello. During this time, he made enough progress (“almost all the orbits of the Collatz map get almost boundary values”, bit.ly/3cHxUVL, 8 September 2019), to generate news outside of mathematical circles, if not yet proven.

Tao’s Collatz mathematics flies some distance over my head, even having some right triangles at a time. For example, these two sentences are typical:

“Theory 1.6 then corresponds to the result of ‘almost global prosperity’, where a large-scale solution needs to be controlled, evolution is close to the boundary state N = O (1). We are inspired by the fact that he demonstrated an almost global wellness result for the local non-linear Schrర్dinger equation, a probability measure for dynamics by combining local wellness theory with an invariant structure. “

However, there is a good resemblance to how he approached the idea of ​​“almost all numbers”. Take a look at a poll conducted before a major election to assess the mood of voters. Clearly, it did not reach every voter — it worked so it had to choose a sample of voters.

For everyone who runs polls, the challenge is: how do you choose such a model? How big should it be? What kind of people should be in it? For example, models that are only for women, or men over the age of 80 or car owners, give completely unreliable results. In other words, you need a model that represents the total electorate. In the same way, Tao wondered if a pattern of positive integers could give him some insight into solving ject han. Say you can compose a reasonable pattern and you will find that most of the numbers in it produce scenes ending with 1. Then it makes sense to think that almost all numbers do the same.

How do you choose a pattern that represents all the numbers nicely? For example, it should not only have powers of 2—4, 8, 16, 512, 2048 and above — because they are again divisible by 2 and quickly reduced to 1. Such a model is trivial, but erroneous, “proof” Similarly, 7 coefficients behave differently to 37, so our sample should have some of the two. Then what happens after you apply the instructions above — the pattern changes its character with each iteration. So you may also want to build your model with this in mind.

In fact, Tao was able to tell how to choose a pattern that, through each iteration, captures its important role. This means that no matter how long the iterations last, the model has a better idea of ​​what is going on and how it will behave.

His conclusion? An article in Quanta Magazine Mentioning that more than 99% of quadrillion numbers produce Collatz sequences indicates that there is a “along the lines” that eventually decreases to less than 200, which is good, because we know that all numbers less than 200 will eventually produce 1.

This is still not complete proof of attendance, and in the end there is no way any proof can proceed. But the same Quanta The article commented: “This is a strong result in the long history of ha.”

In mathematics, it is highly appreciated.

Dilip D’Souza, once a computer scientist, now lives in Mumbai and writes for his dinners. His Twitter handle is DeathEndSfun