Breakthrough in 3n+1 Problem: Collatz Conjecture

I did a study of the Collatz conjecture and found that all even numbers can be removed from the Collatz conjecture because even numbers act as the connecting links between odd numbers.
What do you think about it? Is it a breakthrough in 3n+1 problem?

27 -> 41 -> 31 -> 47 -> 71 -> 107 -> 161 -> 121 -> 91 -> 137 -> 103 -> 155 -> 233 -> 175 -> 263 -> 395 -> 593 -> 445 -> 111 -> 167 -> 251 -> 377 -> 283 -> 425 -> 319 -> 479 -> 719 -> 1079 -> 1619 -> 2429 -> 607 -> 911 -> 1367 -> 2051 -> 3077 -> 769 -> 577 -> 433 -> 325 -> 81 -> 61 -> 15 -> 23 -> 35 -> 53 -> 13 -> 3 -> 5 -> 1.

I like your spirit, but no, it's not a breakthrough. You haven't actually removed any numbers, you've just hidden them from view. To remove them would require that you reformulate the algorithm such that it always avoids even numbers. But 'if even, divide by 2' is literally one of the two rules and the only way to make numbers smaller. Besides, one could just as well say that all odd numbers are just connecting links between even numbers, especially since the goal is to get smaller.

And it's OK to not understand each other in the mathematics. I have some questions.
What is the Collatz conjecture? Is it an algorithm? What kind of algorithm is this?
Is it recursion? What step does the recursion?
If we know the step of the recursion, we can prove the conjecture.

I think this is a breakthrough in 3n+1 problem because the step of the recursion:

[itex]\frac{2n-1}{3}[/itex] for the case n ≡ 2 mod(3),
[itex]\frac{4n-1}{3}[/itex] for the case n ≡ 1 mod(3),
[itex]n=n[/itex] for the case n ≡ 0 mod(3),
and regular using [itex]4x+1[/itex] for new numbers.

To get the number 27, we just need to start the recursion from 1 and use three rules, [itex]\frac{2n-1}{3}[/itex], [itex]\frac{4n-1}{3}[/itex] and [itex]4x+1[/itex].

1 -> 5 -> 3 -> 13 -> 53 -> 35 -> 23 -> 15 -> 61 -> 81 -> 325 -> 433 -> 577 -> 769 -> 3077 -> 2051 -> 1367 -> 911 -> 607 -> 2429 -> 1619 -> 1079 -> 719 -> 479 -> 319 -> 425 -> 283 -> 377 -> 251 -> 167 -> 111 -> 445 -> 593 -> 395 -> 263 -> 175 -> 233 -> 155 -> 103 -> 137 -> 91 -> 121 -> 161 -> 107 -> 71 -> 47 -> 31 -> 41 -> 27.

Martynov_M said:

What is the Collatz conjecture? Is it an algorithm? What kind of algorithm is this?

Well, a conjecture itself is simply an assertion that has no proof as yet.

The Collatz conjecture asserts that - given the algorithm - any n will reduce to 1.

Collatz is an example of a beautifully simple conjecture that current mathematics is unable to resolve. Even though its an example of pure mathematics, it holds a lot of promise for the future techniques that will be invented to solve the conjecture.

Its sequnces are known as hailstone numbers because they rise and fall like hail in a storm before it hits the ground.

There are two ways to prove it:
- find a counterexample ie a cycle where after a time the numbers repeat ie never go to 1
- show that no counterexample exists

Your observation that the even numbers can be reduced to finding it true for odd number has been noted by many people before you which means you are entering the realms of real mathematicians. In the wikipedia article (see below), folks have drawn various diagrams in an attempt to find a pattern that will lead someone to proving the conjecture but so far no luck.

Some other observations have been:
- powers of two all reduce to one using the half a number rule
- there is a statistical observation (geomteric mean) on the odd numbers being 3/4 of the previous odd number in the sequence
- 1 seems to be the only cycle 1->4->2->1...

One caveat, many mathematicians agree that trying to solve the conjecture wholeheartedly will derail your career as a mathematician and so aspiring mathematicians should avoid wasting unnecessary time on it.

Terence Tao has made some significant progress in solving it but he judiciously reserves a small portion of his time to look at these intractable problems and then moves on to more pressing ones.

You can read more about it and related extensions of Collatz on Wikipedia:

https://en.wikipedia.org/wiki/Collatz_conjecture

and this article in Quanta magazine:

https://www.quantamagazine.org/mathematician-proves-huge-result-on-dangerous-problem-20191211/

As an aside, I've often wondered where Collatz could be used and figured it would make a great sequence for slot machines to follow although it may have weaknesses there and pay out too much money.

For example, I had tried the slots in Las Vegas starting with $10 and the game cycle was lose, lose, lose... then win back $9 then lose, lose, lose win back $8... until all my money was gone. It plays to the psychology of the player giving them a hope of a payback but in most cases not really. Sometimes you will see players rush over to a recently vacated machine hoping that luck is only a few turns away.

That kind of cycle is similar to Collatz although the Collatz sequence can soar much higher than the initial starting number and if the player cashes out then the house loses.

Some recent results can be found in this video by the great Terence Tao

I think we can close this thread.

Any "breakthrough" would have been widely acknowledged by now.

jedishrfu said:

Collatz is an example of a beautifully simple conjecture that current mathematics is unable to resolve.

There is little to add. Chances are high that a potential solution will be similar complicated as Wiles's proof of Fermat's theorem.

Thank you for your participation.