Collatz finally solved? [closed]

Question

http://www.newscientist.com/blogs/shortsharpscience/2011/06/simple-number-puzzle-possibly.html

What does mathematicians say of this proof, right or wrong?

Answer 1

Here's a short summary of "what it all boils down to".

Let $k$ be some arbitrary integer. Define a sequence by iterating the mapping $k \mapsto (3k+1)/2$ until you hit an even number. For example, the sequence for $7$ is $$ 7, 11, 17, 26. $$ Denote the even number which terminates the sequence by $F(k)$. So $F(7) = 26$.

We now define a directed graph on all integers $\geq 3$. An edge $j \rightarrow j"$ exists if the sequence for $j"$ contains $F(j)/2$.

For example, we always have $2j \rightarrow j$ and $j \rightarrow F(j)/2$.

Opfer reduces the Collatz conjecture to showing that the vertex $4$ is reachable from any vertex $n \geq 3$. The proof of this fact can be found on page 11.

Note: I obtained this summary by "diagram chasing" across the paper. It's not presented in exactly that way.

Update: On June 17th, 2011 Opfer's pre-print available here has been updated with this comment:

Author's note: The reasoning on p. 11, that "The set of all vertices (2n; l) in all levels will contain all even numbers 2n 6 exactly once." has turned out to be incomplete. Thus, the statement that "the Collatz conjecture is true" has to be withdrawn, at least temporarily.

Answer 2

I don't know whether the paper is right, but it sure looks like a legitimate math paper, and seems to represent the culmination of decades of work by various mathematicians. The author is a reputable mathematician at a major university, and the math genealogy page indicates that he was originally a student of Lothar Collatz.

Apparently the preprint has been submitted to Mathematics of Computation, so it will now undergo the formal peer review process. It is much too early to tell how this will turn out.