Related papers: Collatz Problem of Positive Integers

This paper has been withdrawn by the author due to a crucial error.

This paper has been withdrawn by the author.

This paper has been withdrawn by the author, due to an error in the proof of Theorem 3.8.

This paper has been withdrawn by the author, due a critical mistake on page 3.

This paper has been withdrawn by the author. There is an error on page 3 in the last inequality before Lemma 1.1.

This paper has been withdrawn by the authors due to need of essential revision.

This paper has been withdrawn by the author due to a crucial sign error in equation 1.

This paper has been withdrawn by the author.

This paper has been withdrawn by the author, due a crucial error in the main idea.

This paper has been withdrawn by the author since the proof of Lemma 8 is not correct.

This paper has been withdrawn by the author(s), due a crucial sign error in Thm. 11.

This paper has been withdrawn by the author due to a gap in the proof of Lemma 3.4

This paper has been withdrawn by the author due to an error in section 7. There is a new version: arXiv:1011.3352.

This paper has been withdrawn because of serious errors.

This paper has been withdrawn by the authors, due a crucial error in Sec. 3.

This paper has been withdrawn by the author due to the incorrect argument for the security.

This paper has been withdrawn due to crucial errors.

This paper has being withdrawn by the authors due to an error in the conclusion.

This paper has been withdrawn by the author because there are some typos in proofs.

This paper is withdrawn.