A complete proof of the Collatz conjecture

[Guest]

I see that I can't make myself clearer or make any progress in this way. Here I try another approach to explain and I hope it will be more understandable

[dodomaze]

Well, it's good that you cleaned up the notation and made it easier to follow, but claim A2.1 is still unfinished. The sequence generated by [tex]a_0', a_1', ..., a_{nm}'[/tex] begins to take shape and starts with the same [tex]nd_1, nd_2, ...[/tex] as the original; but, again, there is no mention of how the new sequence ends with the same [tex]nd_{nm}[/tex] as the original. Is [tex]a_{nm}' < a_0'[/tex] already? If so, why? Because if it is not, the sequence has not finished yet. For claim A2.1 to be complete, you must prove that [tex]a_{nm}' < a_0'[/tex] after exactly [tex]nd_{nm}[/tex] divisions by 2 from the last multiplication step. (At the moment I would not know how to do that.)

[Guest]

Why bother with a so-called proof...?

Play the lottery instead.

Good luck!

[Guest]

Claim A2.1 is not that [tex]a'_{nm }<a'_{0}[/tex]. It says that if for the initial value [tex]a_{0}[/tex] there is a sequence [tex]nd_{1}...nd_{nm}[/tex] and the series [tex]a_{0}...a_{nm}[/tex], where [tex]a_{0}...a_{nm-1}[/tex] are odd and [tex]a_{nm}[/tex] is an integer, than for the initial value [tex]a'_{0}=a_{0}+k \cdot 2^{nd}[/tex] there is the same sequence and the series [tex]a'_{0}...a'_{nm}[/tex], where [tex]a'_{0}...a'_{nm-1}[/tex] are odd and [tex]a'_{nm}[/tex] is an integer.
The proof that [tex]a'_{nm }<a'_{0}[/tex] appears in the next claim - claim A3.

[dodomaze]

But how do you explain in claim A2.1 that the sequence ends with [tex]nd_{nm}[/tex] ? (Remember the definition of the sequence.)

[Guest]

OK, let me try an example:
If the initial value is 135, the generated sequence can be (1,1,2,3) and the series will be (135,203,305,229,86). It can also be (1,1,2,4) and the series will be (135,203,305,229,43) - all terms must be odd except the last one, which can be any integer. In the first case, nd=7. Since [tex]2^{7 }=128[/tex], the initial value 135+128=263 also generates a sequence (1,1,2,3) and the series will be (263,395,593,445,167).
In the second case, nd=8. Since [tex]2^{8 }=256[/tex], the initial value 135+256=391 also generates a sequence (1,1,2,4) and the series will be (391,587,881,661,124) - all odd, last any integer. That's what claim A2.1 says.

[dodomaze]

Right, but notice that the reason the numbers 135,203,305,229,86 generated the sequence 1,1,2,3, ending in 3 division steps, is because 86 < 135 - that was your stopping criterion. This is part of the definition of your sequences. And I don't see any reason in claim A2.1 to support your statement that the sequence generated by the starting value 135+128=263 also terminates with 3 division steps. (Please don't detail the steps for the particular case of 263 - claim A2.1 is not about 263 only, is a general claim about an arbitrary sequence constructed from another. The one starting with 263 is only an example; you don't prove this for the general case.)

Suggestion: you docx does not really include the exact definition of a "sequence". Write such a definition, in the form of a list of items that an entity must comply with (like a checklist) in order to be called "a sequence". ("It this a sequence? check... check... check... ok, it fulfills all conditions, so it is a sequence.") Then use this checklist to verify if the object you produce at the end of A2.1 can indeed be called a sequence.

I have to recur to this excruciating level of detail, because you assume something, then forget your assumptions.

[Guest]

Why bother with a so-called proof...?

Play the lottery instead.

Good luck!

Cmon, dynamic duo (Batman & Robin), the truth of ABC conjecture
proves the Beale conjecture, and the Collatz conjecture is also true. Study/review the works on this site.

What's the point of your work?

And why reinvent the "wheels" over again?

Play your hunch (lottery pick) and win a bunch ($$$) if you're so lucky.

Good luck!

[Guest]

I think some of lotteries in the US are rigged.

And the game life is not fair (rigged too). Go figure!

[Guest]

Here I try to define the terms more accurately. Please tell me if now it's correct. TNX

[Nobody Knows]

full proof -> https://www.researchgate.net/publication/351347153_COLLATZ_CONJECTURE_-THE_PROOF
Images that can help to process and understand this proof (they are NOT part of the proof)