Related papers: Collatz Problem of Positive Integers

This paper has been withdrawn by the author, because a better treatment is given in the author's Phd. thesis (Sections 3.4.6 and 4.4), now available on the arxiv.

This paper has been withdrawn

This paper has been withdrawn.

This paper has been withdrawn by the author due to incomplete interpretation for the results.

This paper has been withdrawn due to a critical error discovered in Theorem 4.21. Anyone with a historical or pragamatic interest in prior "negative results", however - e.g., failed proof attempts relating to the (in)consistency of ZF or…

The paper has been withdrawn

This paper has been withdrawn by the author due to a crucial sign error in equation 2.

This paper has been withdrawn by the author due to a mistake in the section 4.

The paper has been withdrawn by the author due to a gap in Proof of Theorem 1.1.

This paper has been withdrawn by the authors.

This paper has been withdrawn by the author due to an error

This paper has been withdrawn by the author due to some error in the result

This paper has been withdrawn by the author(s), due a crucial i-number error in Eqn. 18.

This paper has been withdrawn

This paper has been withdrawn by the author.

This paper has been withdrawn by the authors

This paper has been withdrawn by the author due to a crucial sign error in equation 1.

This paper has been withdrawn by the author due to an error in the proof of Proposition 4.8.

This paper has been withdrawn by the author.

This paper has been temporarily withdrawn by the author because there are errors in it that require further study.