News on proving the Collatz Conjecture

A few weeks ago Gerhard Opfer posted a preprint of a paper titled An analytic approach to the Collatz 3n+1 problem. The paper claims the proof of the Collatz conjecture. In it self that isn’t something really interesting as there are probably several hundred people every year who think they have proven the Collatz conjecture. However, there is one difference here as the paper comes from a research institute. It might sound a little arrogant and imply that non-mathematicians don’t understand math. That is not what I imply, just that the sender packs a bit more punch by being from a research institute.

The Collatz Conjecture

The Collatz Conjecture was originally stated by Lothar Collatz in 1937 and is a problem which is really easy to understand. The conjecture states that

Take an arbitrary natural number and the if  number is even divide by two otherwise multiply by three and add one. If you do this procedure repeatedly the sequence will eventually converge to one.

That is a pretty simple statement that is easy to understand, not unlike Fermat’s Last Theorem.  One example starts with 7 and goes:

7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

As you can also see it jumps up and down quite a bit, but eventually it converges to a number which is the power of two and then sieves through to become one. In Problem 14 of Project Euler we showed that the Collatz Sequence does converge for all numbers up to one million and that the longest sequence  was 525. This was done through a small piece of program and thus by no means a proof for all natural numbers.

To the right you can see one of my cartoons‘s interpretation of Collatz Conjecture.

Thoughts on the paper

I must admit that I have only skimmed the paper.  Gerhard Opfer starts with quoting  a lot of well known work and wants to build on that. I don’t think that this is by any means a bad idea, I just think that with the Collatz Conjecture that has eluded so many people for 74 years by now is very likely to require new breakthroughs or a new branch of mathematics before the problem can be finally solved. It was the case both with Fermat’s Last Theorem and the Poincaré conjecture. This can be seen from the fact that he does not start the paper by presenting new ideas, but jumps directly into technicalities.

It makes me sound a bit jealous which I am not. I really hope that someone proofs it, and for all I care it might very well be the author of this paper. I think it is exciting to follow such breakthroughs in mathematics.

The blogosphere has been buzzing with a discussion of the paper the last few days, and some claim that they have found holes in the proof. I must admit that I am not well versed enough to say if the holes are indeed that. Brent Yorgey has a fairly good explanation of  what the hole in the prof seems to be. Other people Such as atara_x_ia and Yuval Filmus also comes with arguments that sounds to me like much the same arguments as Brent for why the proof is not valid.

All sources agree that Gerhard Opfer proof relies on Opfer building a tree based on the reverse collatz sequence starting from one. And he states that the tree includes all natural numbers. But he never proofs this statement. This invalidates the proof and it is believed that the task of proving that the tree includes all natural numbers is equally hard to proving the Collatz conjecture.

One curious note is that New Scientist reviewed a lot of long standing mathematical problems and showed that among the already solved problem 50% of them were proven after 53 years. So the Collatz conjecture might just be ripe for picking by now.

Update: As of June 17 Gerhard Opfer has withdrawn his claim as it is apparant from the preprint of his paper. I am of course happy that he acknowlegdes the comments he received from many sources, but on the same I was hoping that he had seen something we hadn’t – it would have been a great achievement. I really wish of all my heart that he will be able to remedy the proof or come up with another approach for the same problem. Keep on fighting.

Posted by Kristian


This is exciting!
Please keep us updated if anything interesting happens!

I realized that some of the thoughts I had made missed the blog, so I have updated the post slightly to include a few missing paragraphs.

Meanwhile, opfer withdraw his claim.

Thanks for letting me know I have made an update of the blog post to reflect the latest update.

Bernhard Hanreich

Here you can see some graphs of the collatzproblem
From my point of view it is clear, that collatz is right. But it is only an analizes and not accepted as proof
I hope you enjoy and understand

Jean-Marie Hachey

BOINC: an approach to the Collatz Conjecture …

Excerpts …

The Collatz Conjecture project does research in mathematics, specifically testing the Collatz Conjecture, also known as 3x+1 or HOTPO (half or triple plus one).
About Collatz Conjecture
Collatz Conjecture is a research project that uses Internet-connected computers to do research in mathematics, specifically testing the Collatz Conjecture also known as 3x+1 or HOTPO (half or triple plus one). You can participate by downloading and running a free program on your computer.

(BOINC: Berkeley Open Infrastructure for Network Computing)

BOINC is supported by the National Science Foundation

How BOINC works

This will be capable of disproving the Collatz conjecture in case they find a counter example, but they will not be able to prove it. However, it might generate data for people to get ideas from.

Gogi Pantsulaia

See http://arxiv.ord/pdf/1502.05602v1.pdf

Can you get any decision by computation concerning with Problem 5.1 ?

Robert Hoff

There’s a proposed proof of the Collatz Conjecture that newly arrived here

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