The Collatz conjecture  concerns sequences  defined as follows: start with any positive integer n. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1.

 

Here's a directed graph showing the orbits of small numbers under the Collatz map,  skipping even numbers. 

Anyway,  mathematicians keep saying this is beyond the reach of present day mathematics.   Wonder if humans will solve this about the time we get nuclear fusion to be a practical power source.  

Why the timing coinciding with the advent of fusion? What makes you wonder that? 

Edited November 7, 20214 yr by StringJunky

Commercial fusion was just meant as example of something a ways off in the future.  Note to self:  do not post and try to listen to spouse at same time.   (thanks for noticing the confusingness) 

(deleted, I'm just tired.)

Edited November 8, 20214 yr by pzkpfw

17 hours ago, TheVat said:

Anyway,  mathematicians keep saying this is beyond the reach of present day mathematics.   Wonder if humans will solve this about the time we get nuclear fusion to be a practical power source. 

Interesting...

By definition, any conjecture that stands unproven is beyond the reach of present day mathematics. So Erdos is being kind of tautological, whether on purpose or not, I don't know.

And as to the second sentence, you're comparing two unponderables:

1) The time it will take humans to solve Collatz's conjecture

2) The time it will take humans to get nuclear fusion to be practical ('practical' = ?)

 

I'd say there's no answer to your question ('I wonder...'). Unless 'there's no answer to your question' is considered to be an answer to your question'.

And that, assuming you're asking a question --maybe you're just projecting a wondering.

Full disclosure, this was one of those posts where the OP really just saw something cool and wants to share it, but this website's rules seem to require you say something, advance some sort of opinion, conjecture, speculation, whatever.  So I padded a bit.  But really, I posted hoping that someone with far more of a math mind than I have would offer some interesting thoughts on why or why not a problem can be said to be beyond the reach of present mathematics. 

I know there are problems, like the four color theorem, where a solution only happened with some computer assistance, and so 18th century mathies with paper and quills would not have been able to solve it.  But now, and this I do wonder, it is harder for a math layperson like me to see why something gets ruled as "beyond us." 

Ah.

Seems like by groping in this relative darkness,  you've come up with a very interesting question. I'll let Andrew Wiles do the talking, because I think he's got something to say that's related to your question:

Quote

Andrew Wiles:

Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it’s dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it’s all illuminated. You can see exactly where you were.

[Quoted from Did earlier thoughts inspire Grothendieck? by Frans Oort, who refers to the BBC documentary by S. Singh and John Lynch: Fermat’s Last Theorem. Horizon, BBC 1996.]

[With thanks to Thomas Riepe]

Taken from:

https://micromath.wordpress.com/2011/11/06/andrew-wiles-on-doing-mathematics/

I'd read this quote before as referred to number theory, which Collatz's conjecture is really about. The idea is that you could be microns away from the switch and you wouldn't know. So I'm very skeptic of any relevance of any statement on how close you are to a proof of something. Seems like something that cannot be graded. But that's my feel of it. I have a heartfelt love for pure mathematics, but I don't have the training of a pure mathematician.

Taking up from Wiles' metaphore, I'd ask: Is there a way to judge whether the switch is within your reach before you've touched it? I know I'm waxing metaphorical only, but I think you perfectly understand what I mean.

I remember the late Murray Gell'Mann making similar comments on some 'mathematical people' saying they were 'close to a proof'.

Edited November 8, 20214 yr by joigus
minor correction

On 11/8/2021 at 10:21 PM, joigus said:

I remember the late Murray Gell'Mann making similar comments on some 'mathematical people' saying they were 'close to a proof'.

Here's the quote:

https://books.google.es/books?id=jJzGNl9K5SIC&pg=PA277&lpg=PA277&dq=murray+gell-mann+"close+to+a+proof"&source=bl&ots=Wsndi_9ZiJ&sig=ACfU3U1TVvvcpwOmBDk1GxpNphUqSMmSdQ&hl=en&sa=X&ved=2ahUKEwjNxeDZ-pL0AhVoBGMBHWRADHMQ6AF6BAgCEAM#v=onepage&q=murray gell-mann "close to a proof"&f=false

https://www.worldscientific.com/worldscibooks/10.1142/7101

Article: Particle Theory, from S-Matrix to Quarks

The quote is:

Quote

This question is still unresolved after thirty years, and is the subject of research
and violent controversy right at this very moment. A lot of «pure people» (pure
mathematical physics types, of whose work I cannot understand a single word) have
been studying the matter, and indicate that they are «close to a proof» that this kind
of thing actually happens, at least in \( \lambda \phi^4 \) theory. Now how does a mathematically
pure person get close to a proof? I do not know. 

(My emphasis.)

Edited November 12, 20214 yr by joigus
minor addition

here is a solution https://vixra.org/abs/2201.0186