Collatz’s conjecture: the problem that makes you waste time for mathematicians

There Collatz conjecture It’s a mathematical problem unresolved That looks like a magic game: choose a random number, if it is odd multiply it by 3 and add 1, if it is equal to halve it, then repeat the procedure many times: do we bet that sooner or later you will reach 1? This conjecture was proposed by Lothar Colletz In 1937 and is still unsolved. In recent years it has become rather popular in the world of social media, but it is also famous in the academic field, so much so that the mathematician Paul Erdős He offered 500 dollars for those who resolve it. Let’s see what the problem consists of, why we call it conjecture and how the Resolution attemptsbut we will also establish one of our own small ones record about it.

What Collatz’s conjecture says

First of all, what is a conjecture? In mathematics a conjecture It is a statement that we think can be true, but for which it is not known No demonstration. In practice, let’s assume that it is true because so the experience tells us, but since no one has managed to find the proof of this truth we remain open to the event that in the end it proves false. Resolving the conjecture means definitively establishing if it is true or false.

There Collatz conjecture It seems apparently extremely simple, yet still knows no demonstration. It works like this, we start with a integer And one wonders if it is equal or odd:

1. If the number is even yes Divide by 2
2. If instead it is shots yes Multiply by 3 and the result and yes add 1

By repeating this “infinite” procedure, the conjecture says that sooner or later we will get to have like Risulted 1. And if we continue further from 1, again we will have 1 as a result. Let’s take an example.

Let’s start with the number 5 And we repeat the procedure with every result we achieve:

• 5 is odd, then according to the rule we multiply it by 3 and add 1 obtaining the result 5 × 3 + 1 = 15 + 1 = 16
• 16 it is equal, so we divide it by 2 obtaining 16 ÷ 2 = 8
• 8 is equal, so continuing we have 8 ÷ 2 = 4
• 4 it is equal, so we have 4 ÷ 2 = 2
• 2 is equal, and here we arrive up to 1, that is 2 ÷ 2

We came to 1, as we conjectured! Or rather, as Collatz has conjectured. If we now try to continue once they arrive at 1:

• 1 is odd, we multiply it by 3 and add 1 obtaining again 4
• From 4 you go to 2, as we have already seen
• From 2 you go to 1, and so on. A cycle has been created, from 1 you don’t run away anymore!

Then you can ask yourself: if we start from a any number Different from 5, will we still fall in this infinite cycle? We still reach 1 and then we don’t run away anymore?

According to Collatz, leaving from a any numbersooner or later, we arrive at the number 1only that nobody has managed to try it: in practice we think it’s true, but we are not 100%sure. However, you can try with any starting number and you will see that you will get to 1. If this were not the case … well compliments! You would have found the very first counter -proof of conjecture and would become famous in the world of mathematics.

Attempts to resolve conjecture

Now that you know the conjecture you can take paper, pen and calculator and try firsthand to verify its truthfulness, but be careful because according to the mathematical Terence Tao – winner of the Fields medal in 2006, a kind of mathematics Nobel – it is

One of the conjectures more “Dangerous”, famous having absorbed enormous quantities of time both by professional and amateur mathematicians

In short, although Collatz’s conjecture actually seems true, not even the best mathematicians in the world have still managed to prove it! So do not be fooled, this conjecture has become rather popular on social media and it can happen to come across users of the various social networks who claim to have managed to solve it, but at the moment nobody has not yet succeeded.

Many have tried and the mathematician Shizuo Kakutaniin 1960, stated that for about a month atUniversity of Yale Everyone worked on conjecture, without results. The same happened toUniversity of ChicagoSo that he began to circulate, as a joke, the idea that Collatz’s conjecture was part of a conspiracy to slow down the work of the US mathematicians.

Resolution attempts: look for the error

Among the attempts to solve the conjecture it is worth mentioning the empirical evidence in which, rather than looking for a rigorous demonstration, the existence of a number that – sooner or later – invalid the conjecture. In fact, it would be enough to find only one number for which, reiterating the procedure we have described, you never get to number 1.

To datethe conjecture was found valid For all numbers up to 2,361.183.241.434.822.606.848 (i.e. 2⁷¹ ), so if you intend to look for numbers for which it does not work you should start from higher numbers than this. We did it for the next number, that is 2⁷¹+1 = 2.361.183.241.434.822.606.849: After only 3 steps by applying the rule of conjecture, the result drops below 2⁷¹so we can say that even the next number Check the conjecture.

Obviously nobody could ever be able to test the conjecture on everyone Possible and imaginable numbers, given that they are infinite, so if we want to demonstrate that conjecture is true you have to follow other paths, but it is not so easy, as sanctioned by Paul Erdos in 1983 who said

Mathematics is not yet mature due to problems of this type.

Some decades have passed and again we don’t know how to solve completely this problem despite study of various kinds, including that of Conway who created a programming language, Frattran, based on the rules of Collatz conjecture with which you can perform any operation can be performed with a computer.

The most recent results are those of Tao that, improving the work of other mathematicians who preceded it, was able to demonstrate that the conjecture is true for “almost all” numbers (something like 99.99% of numbers) … but almost all numbers are not all numbers.