## The Collatz Conjecture – recent development?

On his site, John D. Cook recently proliferated a paper by Gerhard Opfer that claimed to solve the Collatz Conjecture. The Collatz Conjecture is simple to state:

Collatz (or the 3n + 1 conjecture):

Starting at any number do the following: if n is even, divide by 2; if n is odd, multiply by 3 and add 1.

The conjecture states that no matter what positive integer you start at, you will end up at 1 (the so-called 1-4-2 loop).

At first, I had high hopes for the paper. It seems relatively well-written and was submitted to the Mathematics of Computation, a very respectable journal. I even sent out a brief email about the paper. But the paper is flawed. The problem, I think, can be succinctly summarized by the following: he relies on the assumption that starting with any number $latex n_0$, one will eventually hit a number that is less than $latex n_0$. When stated like this, it seems obvious that there is a problem, but he only relied on that one number (rather than the apparent infinite descent that could follow). The exact problem occurs with his ‘annihilation argument’ on page 11 of the pdf above. He more or less states that one can start at 1 and reach every number by doing a sort of reverse Collatz function (he’s actually a bit wittier than that), but does not prove it.

More commentary can be found on reddit, reddit again, and on math.SE (a question protected by Qiaocho Yuan – go him).

I use this as an intro to a sort of joke that goes around mathematician’s circles. A while back, Sean Carroll wrote up ‘The Alternative-Science Respectability Checklist,’ and it’s *awesome*. Find it here. It turns out that Scott Aaronson wrote up a similar article, inspired by Sean Carroll, that is titled “Ten Signs a Claimed Mathematical Breakthrough is Wrong.”

His inspiration was the time-old problem that simply stated problems encourage generations up people to attack them, and frequently to think that they have made progress. So he asks :

Suppose someone sends you a complicated solution to a famous decades-old math problem, like P vs. NP. How can you decide, in ten minutes or less, whether the solution is worth reading?

And thus his 10 signs were created. I happen to have heard a few people say that this most recent paper on the Collatz Conjecture only failed three: #6 (The paper jumps into technicalities without presenting a new idea), #8 (The paper wastes lots of space on standard material), and #10 (The techniques just seem too wimpy for the problem at hand). {though perhaps #8 is debateable – some say it’s related to a different convention of writing papers, but I don’t know about any of that}

In my experience, I rely mostly on #1 (it’s not written in $latex TeX$), #4 (it conflicts with some impossibility result), and #7 (it doesn’t build on any previous work). But both of these articles are very funny, though not exactly precise nor entirely true.