Property I: From every convergent Collatz sequence, we can form a bounded Collatz set, [tex]C_{even }[/tex], of positive even integers.

Property II: The union of all possible bounded Collatz sets, [tex]C_{even }[/tex], of positive even integers, is the set, E, of all positive even integers.

What are the proofs of properties, I and II?

Hints: The proofs are trivial, but their implications may not be trivial in regards to the truth of the Collatz Conjecture. Please ponder the algorithm for the Collatz Conjecture.

Important Question: Does there exists an exceptional set, [tex]\Tau[/tex], of positive even integers, [tex]\tau[/tex], that violates the Collatz Conjecture?

Hints: Please review the proof of Collatz Conjecture at the link below and refer to the algorithm for the Collatz Conjecture.

And try to find/construct at least one positive even integer that violates the Collatz Conjecture or prove that it exists.

Good luck!

Remarks: You will be famous if you can completely prove or disprove the Collatz Conjecture for all positive even integers. Any half measures are unacceptable! For example, the recent questionable proof of the Collatz Conjecture for almost all positive integers by Terence Tao is unacceptable!

Relevant Reference Links:,

'Collatz Conjecture',

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

'Proof of Collatz Conjecture'

https://www.math10.com/forum/viewtopic.php?f=63&t=1485&start=20.