A Solution of The Collatz Conjecture Problem

baoyuan duan

doi:10.20944/preprints202301.0541.v12

Comment 2

Received: 21 October 2023
Commenter:

baoyuan duan

Commenter's Conflict of Interests: I am one of the author

Comment: According to my conclusion, we can make a bold prediction and verify: In all known convergence odd Collatz sequences, if start odd a multiply 3 plus 1 divided by 2^k(k>=3) get an odd, the corresponding odd of a in (*3+2^m-1)/2^k odd sequence is b, MSB bit of b is 2^n, then the count of all downward steps(k>=3) from a to final 1 is smaller than n! For example, Collatz odd 1893, corresponding odd is 2203, highest bit is 2^11, all downward steps(k>=3) of 1893 is 5<11.

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Response 1 to Comment 2

Received: 23 October 2023
Commenter:

baoyuan duan

Commenter's Conflict of Interests: I am one of the author

Comment: The prediction is a bit adventurous. Best way is to use 3/4 proportional sequence of transform position increment to calculation solution of position inequality, then estimate the count of downward steps and verify.

Response 2 to Comment 2

Received: 24 October 2023
Commenter:

baoyuan duan

Commenter's Conflict of Interests: I am one of the author

Comment:

For example,Collatz odd 174757,actual downward steps(k>=3) is 11.Corresponding odd in (*3+2^m-1)/2^k is 524288-174757=349531, transform position(to itself layer) is (349531+1)/2=174766,odd follow 349531 is (349531*3+524288-1)/16=98305,transform position is 196612, position increment is 196612-174766=21846, position inequality is 174766+21846*4*(1-0.75^(n-1))>262144, solution is n>34. 11<34, satisfies conclusion in paper.According to my conclusion,almost all known convergence odd Collatz sequences have the same principle(huge odd is not a strong precondition in paper).

Response 3 to Comment 2

Received: 25 October 2023
Commenter:

baoyuan duan

Commenter's Conflict of Interests: I am one of the author

Comment: In this example,n represents transform position serial number.If we want it represent step serial number(in order to compare), should use inequality 174766+21846*4*(1-0.75^n)>262144, same as established, as long as actual convergence steps>2 and first k>=3. I have verified up to 2^23-3 using computer, no exception cases exist.

Comment 1

Received: 17 October 2023
Commenter: baoyuan duan
Commenter's Conflict of Interests: Author

Comment: Correct some errors and make the manuscript shorter. Key proof is in section 6. Although maybe still exists some non-Key errors(i have not much time to checked), i believe the proof is correct. Because the logic and calculation are not complex, main contribution is the idea.

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A Solution of The Collatz Conjecture Problem

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