A few days ago John Cook reported a draft paper claiming to solve the Collatz conjecture. Of course, since the Collatz conjecture is so simple to state, it constantly attracts tons of would-be solvers, and most of the purported “proofs” they generate are not even worth mathematicians’ time to look at. So why should this one be different? Well, on the surface, it seemed to be a much more serious attempt than the vast majority. Of Scott Aaronson’s Ten Signs a Claimed Mathematical Breakthrough is Wrong, this paper exhibited only #6 (jumping straight into technical material without presenting a new idea) and #10 (using techniques that seem too wimpy). But #6 could just be due to poor organization of the paper, and #10 is not obvious until you get to the end.
But — and you knew this was coming — it is wrong. I’ve spent several hours over the past few days reading over the paper and came to this conclusion independenly — and then found several other people who had come to the same conclusion, for the same reason.
Consider the following “proof” of the Collatz conjecture. We consider running the Collatz function (call it ) “backwards”, to find out which number(s) could have preceded a given number in a Collatz sequence. We always have , so from we can go backwards to . Also, when is an odd integer, that is, when is one more than an odd multiple of . These are the only two possibilities. So, starting from , we can go backwards to ; from we can go backwards to ; from we can go to or ; from we get ; from we get or ; and so on. In this way we can build up an infinite tree of numbers. But this tree contains every natural number since we can always work up the tree from until we find any number we want. Hence every number must reach when iterating the Collatz function.
Do you see the problem with this “proof”? Well, this is essentially what Gerhard Opfer’s “proof” boils down to as well (the details are not exactly the same, but the form is pretty much identical). There is a lot of stuff first about linear operators over complex functions, but for this part he is just relying on work that someone else already did anyway, and in the end he just ends up looking at coefficients of power series, which just boils down to some number theory. He builds a tree by inverting a certain function (not the Collatz function but a closely related one), and the whole argument rests on the fact that this tree contains every natural number — which he states but does not prove! And it seems clear to me that proving this would be no easier than proving the Collatz conjecture itself. In the end, for all the detailed argument and contortion, he has come back precisely to where he started.
(By the way, I am not interested in reading your supposed proof of the Collatz conjecture, so please do not post a link to it in the comments!)