71–80 of 102 posts
Is there something wrong with that proof? https://www.youtube.com/watch?v=O7SmNyte6nA (in French) The 3x+1 Conjecture is false Text in English : https://arxiv.org/abs/2104.10681
I watched this the other day. I was left wondering one big question. Why does this need to be solved? Would that even change anything?
Would it change anything? Who knows—we have yet to solve it using our current mathematical knowledge, so who knows what new mathematical knowledge will be developed in order to solve it?
Is there something wrong with that proof? https://www.youtube.com/watch?v=O7SmNyte6nA (in French) The 3x+1 Conjecture is false Text in English : https://arxiv.org/abs/2104.10681
This statement is not proven, and is not supported by the previous theorems. Thus the following statement "We therefore conclude that the 3x+ 1 conjecture is false" remains an unproven assertion.
Essentially the paper argues that the density F(k) of trajectories at least as long as 'k' gets ever smaller as k increases, without ever reaching zero, and thus there exists at least one value that has an infinity long trajectory.
But that's as false assertion. Just because F(k) is non-zero for every finite value of k does NOT imply that the lim as k tends to infinity of F(k) is non-zero, and consequently does not prove the existence of an infinity long trajectory.
TL;DR: paper applies intuition on finite sequences to infinite sequences and gets it wrong.
I recommend Veritasium, his videos are quite good. This one probably has good and useful visualizations but I was doing the cleaning while listening to it so I missed that. Still possible to follow without watching though.
Absolutely. His "gravity is not a force" video led me through a whole range of emotions from "this is ridiculous" to "actually..." to "holy crap, I've been wrong about this my entire life". In terms of sheer brain-breaking... this is my favourite: https://www.youtube.com/watch?v=ovJcsL7vyrk But this is the one I think people on HN need to watch most https://www.youtube.com/watch?v=3LopI4YeC4I
I watched this the other day. I was left wondering one big question. Why does this need to be solved? Would that even change anything?
> Why does this need to be solved? Obviously it "doesn't". Equally obviously this is mathematics, and it's worth doing if it's interesting. But a somewhat more serious answer is this: the fact that we don't know the answer to this relatively simple question implies strongly that we don't know how numbers work . And if there are obvious gaps in our understanding of number theory that show themselves in trivial ways li…
Earlier quoted context omitted.
The conjectures where the condition is very simple (Collatz, Goldbach, Fermat) but the proof is very elusive are not as common as you think.
I guess even if they're not common, clearly the person who first came up with 3x+1 didn't know it was going to be so hard to solve. So it was a random simple problem to formulate that ended up being very difficult to solve.
This was a great video. But the one titled "Math Has a Fatal Flaw"[1] was even more interesting: the meta-mathematics, thus mathematics of entire mathematics (Gödel). [1] https://www.youtube.com/watch?v=HeQX2HjkcNo
I can't get over the clickbaity, obviously incorrect title to that one. I refuse to watch it for that reason alone. If the flaw is fatal, then every mathematician is wasting their time.
My biggest take-away from this was FRACTRAN[1], The Bestest Ever™ programming language designed by the (sadly, late) John Conway. To run a FRACTRAN program, you lookup its catalogue number, and repeatedly evaluate a certain simple function on it (which has the same spirit as the 3x + 1 one in the video). As in 3x+1, all operations are integer operations. FRACTRAN is Turing-complete, of course, so you can rewrite any…
> 3x+1 is, too, a programming language What does this mean? Isn't having a catalog essentially precomputing a function for every possible input?
A 3x + 1 program is a positive integer N. "Running" this program would be iterating 3x + 1 until it gets stuck in a loop (this would be the "halting condition"). The "output" would be a fixed function of the numbers in the loop (e.g. the loop itself; like 4, 2, 1).
As far as we know, all "programs" in 3x + 1 output 4, 2, 1. This makes for a very boring language (essentially, one you can only make "Hello World!" in). But we don't have a proof that that's all there is to it.
>Isn't having a catalog essentially precomputing a function for every possible input?
Specifically, that means that given a Turing machine T and input X, we can compute a number N = N(T, X) such that the "output" of running our "language" on N is the same as the output of T on X.
Which is what they do in FRACTRAN: they show how to encode an arbitrary program and its input as a number N, and how to read the output from the iteration.
Earlier quoted context omitted.
If 3x+1 is a programming language, the Collatz conjecture must be false as otherwise it would always halt. I suspect 3x+1 is not Turing complete.
Or maybe it is, and Collatz conjecture is false.
Though honestly, I really wouldn't bet on ever getting anything other than the 4 - 2 - 1 loop out of it in the end.