Related papers: Collatz Problem of Positive Integers
This paper has been withdrawn by the author, due to errors in the figures.
This paper has been withdrawn by the author due to crucial errors.
This paper has been withdrawn by the author due to a crucial sign error in equation 1.
This paper has been withdrawn by the author due to an error in the proof of Theorem 1.
The paper is withdrawn.
This paper has been withdrawn by the author due to serious flaws in certain proofs. For instance, the method used to construct certain automorphic representations is flawed.
We have observed a gap in one of the arguments in the main theorem. We choose to withdraw the paper until we rectify the gap.
This paper has been withdrawn by the authors due to a gap in the proof of the main result (in 5.3).
The paper has been withdrawn due to an error in the main theorem.
This paper has been withdrawn by the author due to crucial sign error in Lemma 8.1.
withdrawed due to a substantial error.
The paper has been withdrawn by the author due to crucial error in formula (14) which does not satisfy conditions for S(x,y) domain if only d>2.
This paper has been withdrawn Abstract: This paper has been withdrawn by the author due to the publication.
This paper has been retracted.
This paper has been withdrawn by the author due to a errors in figure 3,4
The paper is withdrawn by the author due to an oversimplified and misleading approach which was taken initially as a starting point.
This paper has been withdrawn by the author(s), due a crucial sign error in Eqn. 1.
This paper has been withdrawn by the author due to an error in the computation of E(n,x) on page 6 which appears to be essential for the result. The author is currently trying to correct this proof
This paper has been withdrawn because Proposition 2.2 (c) is false. This invalids the main results of section 2 and 3. We thank A. Canonaco for pointing us the error.
This paper has been withdrawn by the author, due to an error in the inequality after the inequality (31).