A Solution of The Collatz Conjecture Problem

baoyuan duan

doi:10.20944/preprints202301.0541.v7

Comment 5

Received: 2 June 2023
Commenter:

baoyuan duan

The commenter has declared there is no conflict of interests.

Comment: Maybe someone worry about the backward ratio become smaller than 3/4 after we merge previous steps,it is not worth worrying about,we can merge any backward steps to one step,not limit to property of 1 or 2. Suppose exist a nonconvergence long huge sequence, start number property 2^p is bigger than 4, after n steps upward k layers(exist upward,forward and downward), main trend of k must be increased and must also be infinited if n is infinited,and the final transform position must be bigger than the case: from the second step the number property 2^q is 2,and the total steps number is k+p-2(also upward k layers).in the second case,through merge backward steps,we can easily get a position sequence which average ratio is bigger than 3/4 through merge backward steps,and then,the final position must be bigger than 2^(m-1) if k is infinited(new steps number after merging is also infinited).

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Response 1 to Comment 5

Received: 2 June 2023
Commenter:

baoyuan duan

The commenter has declared there is no conflict of interests.

Comment: k+1+p-2

Response 2 to Comment 5

Received: 3 June 2023
Commenter:

baoyuan duan

The commenter has declared there is no conflict of interests.

Comment: This is considered only from transform position view. In actual world,backward steps are not possible to exist infinited successive upward steps,and can not converge only through upward steps

Response 3 to Comment 5

Received: 4 June 2023
Commenter:

baoyuan duan

The commenter has declared there is no conflict of interests.

Comment: pen error,should be behind steps, not backward steps.

Response 4 to Comment 5

Received: 9 June 2023
Commenter:

baoyuan duan

The commenter has declared there is no conflict of interests.

Comment: In fact, even the second case sequence has the same stepcount n with the origional sequence, its final transform position is still smaller than the final transform position of original sequence, bacause, to any specific odd, upward step has the smallest transform position increment.

Comment 1

Received: 11 April 2023
Commenter: baoyuan duan
Commenter's Conflict of Interests: Author

Comment: make the proof more detailed and stronger

Comment 2

Received: 15 April 2023
Commenter:

Comment: If needed, we can also think successive (2,2) as one step, this further guarantees the average ratio of long huge sequence is >3/4.

Comment 3

Received: 21 April 2023
Commenter:

Comment: Sorry, there is a pen error in page 2. The number on the arrow of 425->319 should be 2, not 1.

Comment 4

Received: 10 May 2023
Commenter:

Comment: Sorry, there is a pen error in page 15, "this serial number could be possible to
appear frequently" should be "this serial number could not be possible to
appear frequently"

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A Solution of The Collatz Conjecture Problem

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