5 ms·

What is Collatz conjecture good for?

3 points by zekchelovek 2y ago

Can anyone tell me what real world problem is solved by proving Collatz conjecture? Outside of (maybe) being awarded $800k years down the road?

pkoird 2y ago
The conjecture itself might not be as important but the tools we have to discover and use to prove this conjecture would be of paramount importance, as they'd teach us about the nature of mathematics, numbers, and more.
- zekchelovek 2y ago
  That sounds about like every other answer i get about it. Not much incentive to work on it.
  - rgovostes 2y ago
    What was the value in the moon landing?
    - zekchelovek 2y ago
      There was a cheese shortage. Everyone knows this
mikewarot 2y ago
It's a curiosity, a rabbit hole, and not likely to have practical applications. It's one of the math things I toy with from time to time, like factoring large coprimes.
- wruza 2y ago
  To me it’s hard to believe that fundamental knowledge about computation would have no practical applications. That depends on “practical” of course. Elliptic curves or group theory were hardly of any use for medieval dirt slappers.
  - zekchelovek 2y ago
    I made the mistake of finding answers to the hardest useful problems first (factoring, Goldbach's), and now I'm looking for aomething else to work on.
- zekchelovek 2y ago
  Already done with factoring large coprimes. Looking for next puzzle to work on. Can't get overly interested in useless puzzles (like sudoku, etc).
sophiebits 2y ago
Many unsolved problems in mathematics are ones that are not easily understood by a lay person. The Collatz conjecture has captured hearts and minds at least in part because it is easy to understand the statement of. Perhaps proving that pi is normal would be another one. But they are few and far between. In a sense, the reason it’s interesting is that we don’t know how to solve it.
EVa5I7bHFq9mnYK 2y ago
Well, real world problems, like fluid dynamics or machine learning, are too hard for mathematicians to tackle, so they resort to toy problems.
7373737373 2y ago
The https://en.wikipedia.org/wiki/Busy_beaver https://en.wikipedia.org/wiki/Busy_beaver value for machines with 2 symbols and 5 states (the maximum number of steps that such a machine can make on an initially blank tape before eventually halting) was recently proved to be 47,176,870: https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-proved-bb-5-47-176-870/237 https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-prov... Finding the value for machines with 2 symbols and 6 (or more) states will require solving, among other things, the problem of whether the following Collatz-like program ("Antihydra": https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html, https://wiki.bbchallenge.org/wiki/Cryptids https://wiki.bbchallenge.org/wiki/Cryptids, https://wiki.bbchallenge.org/wiki/Antihydra https://wiki.bbchallenge.org/wiki/Antihydra) halts or not, because there exists such a machine that exhibits this behavior: a = 8 b = 3 while b: a += a>>1 b += 2-(a%2)*3 This looks simple, but no one expects it to be solved in the near future. So Collatz(-like) problems are intimately connected to the halting problem and in this sense represent the "simplest" type of problem (statement) in the Cryptid/complexity hierarchy we cannot (yet?) solve. Other ((closer to the) real world) problems are higher in this hierarchy (require more states to describe: https://wiki.bbchallenge.org/wiki/Cryptids#Larger_Cryptids https://wiki.bbchallenge.org/wiki/Cryptids#Larger_Cryptids), and solutions/increased understanding of the Collatz problems could assist in their solution.
- zekchelovek 2y ago
  Thank you for giving me an answer that nobody else seems to know. I have no idea if you're correct, since i haven't studied the halting problem, although i understand the general concept.

3 points by zekchelovek 2y ago

Can anyone tell me what real world problem is solved by proving Collatz conjecture? Outside of (maybe) being awarded $800k years down the road?

pkoird 2y ago

The conjecture itself might not be as important but the tools we have to discover and use to prove this conjecture would be of paramount importance, as they'd teach us about the nature of mathematics, numbers, and more.

zekchelovek 2y ago
That sounds about like every other answer i get about it. Not much incentive to work on it.
- rgovostes 2y ago
  What was the value in the moon landing?
  - zekchelovek 2y ago
    There was a cheese shortage. Everyone knows this

mikewarot 2y ago

It's a curiosity, a rabbit hole, and not likely to have practical applications. It's one of the math things I toy with from time to time, like factoring large coprimes.

wruza 2y ago
To me it’s hard to believe that fundamental knowledge about computation would have no practical applications. That depends on “practical” of course. Elliptic curves or group theory were hardly of any use for medieval dirt slappers.
- zekchelovek 2y ago
  I made the mistake of finding answers to the hardest useful problems first (factoring, Goldbach's), and now I'm looking for aomething else to work on.
zekchelovek 2y ago
Already done with factoring large coprimes. Looking for next puzzle to work on. Can't get overly interested in useless puzzles (like sudoku, etc).

sophiebits 2y ago

Many unsolved problems in mathematics are ones that are not easily understood by a lay person. The Collatz conjecture has captured hearts and minds at least in part because it is easy to understand the statement of. Perhaps proving that pi is normal would be another one. But they are few and far between. In a sense, the reason it’s interesting is that we don’t know how to solve it.

EVa5I7bHFq9mnYK 2y ago

Well, real world problems, like fluid dynamics or machine learning, are too hard for mathematicians to tackle, so they resort to toy problems.

7373737373 2y ago

The https://en.wikipedia.org/wiki/Busy_beaver https://en.wikipedia.org/wiki/Busy_beaver value for machines with 2 symbols and 5 states (the maximum number of steps that such a machine can make on an initially blank tape before eventually halting) was recently proved to be 47,176,870: https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-proved-bb-5-47-176-870/237 https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-prov... Finding the value for machines with 2 symbols and 6 (or more) states will require solving, among other things, the problem of whether the following Collatz-like program ("Antihydra": https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html, https://wiki.bbchallenge.org/wiki/Cryptids https://wiki.bbchallenge.org/wiki/Cryptids, https://wiki.bbchallenge.org/wiki/Antihydra https://wiki.bbchallenge.org/wiki/Antihydra) halts or not, because there exists such a machine that exhibits this behavior: a = 8 b = 3 while b: a += a>>1 b += 2-(a%2)*3 This looks simple, but no one expects it to be solved in the near future. So Collatz(-like) problems are intimately connected to the halting problem and in this sense represent the "simplest" type of problem (statement) in the Cryptid/complexity hierarchy we cannot (yet?) solve. Other ((closer to the) real world) problems are higher in this hierarchy (require more states to describe: https://wiki.bbchallenge.org/wiki/Cryptids#Larger_Cryptids https://wiki.bbchallenge.org/wiki/Cryptids#Larger_Cryptids), and solutions/increased understanding of the Collatz problems could assist in their solution.

zekchelovek 2y ago
Thank you for giving me an answer that nobody else seems to know. I have no idea if you're correct, since i haven't studied the halting problem, although i understand the general concept.