Veritasium Does the Collatz Conjecture

 After the usual routine introduction to the Collatz Conjecture, Veritasium plunges on with this wonderful recent exploration of it, enjoyable by young and old:

https://www.youtube.com/watch?v=094y1Z2wpJg

Collatz Conjecture... perhaps a new approach

 For any in the mood for some heavier reading here's "An Automated Approach To the Collatz Conjecture" newly out from Scott Aaronson et.al.:

https://arxiv.org/pdf/2105.14697.pdf

Collatz… So what’s the history of it???

I see the always-intriguing Collatz conjecture going around a bit again on Twitter (as it seems to every few months), but just started wondering what the history/background of it is, which I’ve never seen much about, other than that it originated with Lothar Collatz maybe in the 1930s(?).

The simple statement of it, is that you take any positive integer and apply the following 2 rules iteratively:

• If the number is even, divide it by two, or
• If the number is odd, triple it and add one. (Then repeat.)

Doing so successively you will always conclude with a sequence of integers ending at 4, 2, 1 (...or so goes the conjecture).

People write a lot about the conjecture and continue to work on it, but what I’m wondering now is how did Collatz stumble upon those two specific iterative rules to begin with out of essentially an infinite number that might be imagined (even if many would pretty obviously not lead to anything interesting)? Or, you could even come up with 3 iterative rules! Or, or, or… Did he try LOTS of others… have other people since tried LOTS of others? Is there something unique about his two rules, as opposed to ANY others that might be concocted and have some interesting result?

Anyone know, or can point to some informative links?

...And for anyone who's missed it, here's a nice Numberphile introduction to the Collatz conjecture:

ADDENDUM:

In the comments below Brian Hayes responds with this link to an old piece he wrote for Scientific American on the subject. Like other pieces, it’s largely analysis of the conjecture, written in Brian’s always-superb exposition, but there is a bit of history on page 12. He also references a piece by Lothar himself, but what I found most interesting in tracking it down, was seeing a number of folks say that though Lothar explored many iterative functions, he never actually claimed specific credit for the so-called 3N+1 problem that took on his own name!

And with all that said, what I’m still not clear about is whether the two conjecture rules involved in 3N+1 were arrived at primarily by sheer trial-and-error, or was there a more methodological/quantitative approach to hitting upon them?

Conway, Collatz, Chaos

19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

Ivars Peterson discusses a recent "provocative" article from the always-interesting John Conway on the provability of various mathematical claims, with a focus on the Collatz conjecture (one of those famous easy-to-state, difficult-perhaps-impossible-to-prove conjectures):

http://mathtourist.blogspot.com/2013/03/wild-beasts-around-corner.html

It starts off thusly:

"Some mathematical problems are easy to describe but turn out to be notoriously difficult to solve. In some instances, these difficulties may stem from fundamental issues of provability, especially for mathematical problems apparently poised between order and chaos."

Even xkcd has taken notice of the Collatz conjecture.

Collatz Fans

If you're really into the Collatz (3n+1) Conjecture nice new long piece on it here:

http://home.graysoncable.com/dkcox/

Stuff For a Tuesday

I previously posted about the intriguing and unsolved "Collatz Problem" (whether or not certain created sequences always must end in the same pattern, regardless of starting point). It is also known by the name "Hailstone Numbers" and Ben Vitale recently wrote about them here:

http://benvitale-funwithnum3ers.blogspot.com/2010/10/list-of-squares.html?spref=tw

Clifford Pickover addresses the same subject here:

http://sprott.physics.wisc.edu/pickover/hailstone.html

Meanwhile, in a different vein, I just recently discovered this fairly young blog devoted entirely to prime numbers:

http://primepatterns.wordpress.com/

Finally, I've never been much of a Sudoku fan, but I do very much enjoy Ken-Ken (are there others like me out there, and if so, why is that???)... In any event, this poster's been thinking more about Ken-Ken than I ever did:

http://bit-player.org/2010/kenken-friendly-numbers

Collatz Conjecture

A conjecture that is easy to state, easy to understand... but, difficult to prove:

http://jpcameron.com/blog/?p=47

http://en.wikipedia.org/wiki/Collatz_conjecture