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 $ n_0$, one will eventually hit a number that is less than $ 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 $\TeX$), #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.

### Leave a comment

## Info on how to comment

To make a comment, please send an email using the button below. Your email
address **won't be shared** (unless you include it in the body
of your comment). If you don't want your real name to be used next to your
comment, please specify the name you would like to use. If you want your name
to link to a particular url, include that as well.

**bold**, *italics*, and plain text are allowed in
comments. A reasonable subset of markdown is supported, including lists,
links, and fenced code blocks. In addition, math can be formatted using
`$(inline math)$`

or `$$(your display equation)$$`

.

**Please use plaintext email** when commenting. See
Plaintext Email and
Comments on this site for more. Note also that
**comments are expected to be open, considerate, and
respectful.**

Comments (1)2011-06-06 davidlowrydudaPlease note that I moderate first-time posters on my blog. There have already been a new proof posted within the space left for comments here. I will not let those through.

If, however, you would like to attempt to give such a proof to the world, I recommend posting it on Math.stackexchange, posting it or mymathforum.com, or sending it to a mathematician who will read it. Or $\TeX$ it up and find someone who will authorize you on the arxiv.

(Edited later: the comment system has been completely rewritten on this site).