Related papers: Collatz Problem of Positive Integers

This paper was withdrawn by the authors.

This paper has been withdrawn by the authors due to a crucial error.

This paper has been withdrawn by the authors due to crucial error in the main proof (located in Section 2.4). The authors apologize for any inconveniences.

This paper has been withdrawn by the authors due to some fatal errors in the analysis.

This paper has been withdrawn by the author due to a crucial error in one of equation in (3).

This paper has been withdrawn due to a critical error near equation (71). This error causes the entire argument of the paper to collapse. Emmanuel Candes of Stanford discovered the error, and has suggested a correct analysis, which will be…

The paper has been withdrawn due to numerical error.

This paper has been withdrawn due to a error in Theorem 3.1.

This paper has been withdrawn by the author due to a crucial sign error.

This paper has been withdrawn by the author, due to a crucial error in the proof of Thm.1

This paper has been withdrawn by the author, due a crucial error in the optimization. For the last one month I have been trying to remove the error, but it seems to take a lot of time so I decided to withdraw this paper for the moment.

This paper has been withdrawn by the authors due to an error in the main theorem.

This paper has been withdrawn by the author because the conclusions reached in it are incorrect.

This paper has been withdrawn.

This paper has been withdrawn by the author due to an error in the proof of Theorem 6.

This paper has been withdrawn by the author, due a crucial error in Eq. 6.

This paper has been withdrawn by the author due to errors.

This paper has been withdrawn by the author(s), due the final version in math.QA/0604564

This paper has been withdrawn by the author due to a crucial sign error in Proposition 3.1.

This paper has been withdrawn by the author