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The seventh most popular easily understood unsolved problem on MathOverflow

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Re: The seventh most popular easily understood unsolved problem on MathOverflow

#31

Post by cubefox »

eutectic wrote: post #20
Buttons840 wrote: post #8

One of the comments was "the Collatz conjecture feels like we're missing a branch of mathematics", or something to that effect. I followed that rabbit hole a bit and found the Plya conjecture[0], which was disproven when a counter example was found at approximately 10^361. Here I was naively thinking that if no counter examples were found in the first, say, 10^10 numbers, no counter examples should exist. If only it…

John Conway showed that if you generalise the coefficients of the Collatz conjecture then some instances are undecidable. So maybe there is no proof.

If it is "undecidable" this means there is no counterexample to the Collatz conjecture, since any counterexample would disprove it. But the Collatz conjecture does exactly state that there are no counterexamples. Which means: If it is undecidable, it is true.

Which seems a bit paradoxical. If you can prove that the Collatz conjecture is undecidable, you would also prove that it has no counterexamples, and thus that it is true. Which would make it decidable -- contradiction. So this seems to prove that if the Collatz conjecture is undecidable, this fact is itself also undecidable.

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#32

Post by zimpenfish »

permo-w wrote: post #23
Buttons840 wrote: post #8

One of the comments was "the Collatz conjecture feels like we're missing a branch of mathematics", or something to that effect. I followed that rabbit hole a bit and found the Plya conjecture[0], which was disproven when a counter example was found at approximately 10^361. Here I was naively thinking that if no counter examples were found in the first, say, 10^10 numbers, no counter examples should exist. If only it…

>Here I was naively thinking that if no counter examples were found in the first, say, 10^10 numbers, no counter examples should exist why on earth would you ever think that

> naively

I think that answers your question?

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#34

Post by dooglius »

cubefox wrote: post #31
eutectic wrote: post #20

Earlier quoted context omitted.

John Conway showed that if you generalise the coefficients of the Collatz conjecture then some instances are undecidable. So maybe there is no proof.

If it is "undecidable" this means there is no counterexample to the Collatz conjecture, since any counterexample would disprove it. But the Collatz conjecture does exactly state that there are no counterexamples. Which means: If it is undecidable, it is true. Which seems a bit paradoxical. If you can prove that the Collatz conjecture is undecidable, you would also prove that it has no counterexamples, and thus that i…

You missed the key part

> if you generalise the coefficients

The result appears to be https://gwern.net/doc/cs/computable/1972-conway.pdf if you want to read in detail

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#35

Post by [deleted] »

permo-w wrote: post #23
Buttons840 wrote: post #8

One of the comments was "the Collatz conjecture feels like we're missing a branch of mathematics", or something to that effect. I followed that rabbit hole a bit and found the Plya conjecture[0], which was disproven when a counter example was found at approximately 10^361. Here I was naively thinking that if no counter examples were found in the first, say, 10^10 numbers, no counter examples should exist. If only it…

>Here I was naively thinking that if no counter examples were found in the first, say, 10^10 numbers, no counter examples should exist why on earth would you ever think that

[deleted]

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#36

Post by hyperhello »

feoren wrote: post #29
screenoridesaga wrote: post #14

I love how these raw mathematicians consider something proved when they can understand, meanwhile the computer can prove it easily just by counting a finite number of bits. What exactly would be considered proof in this case? Any explanation only mathematicians can understand?

> the computer can prove it easily just by counting a finite number of bits Did you miss the infinite sum there? How would you prove an infinite sum equals a transcendental number by counting finite bits? You'd have to count infinite bits.

Wouldn’t you be able to see the difference converging on zero at least? Unless it oscillates all over but seemed to average. I don’t know if the squeeze theorem applies.

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#37

Post by Sharlin »

datameta wrote: post #18

This may be a question that misses the point - respectfully, what are some practical applications in physics or engineering for such proofs and/or the search for a conjecture counterexample?

None that we know of. We still do it because it's worth doing.

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#38

Post by tzs »

Here's a fairly easily understood problem, which if you could solve it would make you famous in the mathematical world and win you a million dollar prize:

  For a positive integer n:

    Let H(n) = 1 + 1/2 + 1/3 + ... + 1/n

    Let D(n) = the sum of the positive integers that divide n. E.g., D(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28.

  Prove or disprove that for any positive integer n > 1:

    D(n)
That easy to understand problem turns out to be equivalent to the Riemann hypothesis [1], which is one of the most famous and important unsolved problems in number theory.

[1] https://arxiv.org/abs/math/0008177

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#39

Post by thesz »

eutectic wrote: post #20
Buttons840 wrote: post #8

One of the comments was "the Collatz conjecture feels like we're missing a branch of mathematics", or something to that effect. I followed that rabbit hole a bit and found the Plya conjecture[0], which was disproven when a counter example was found at approximately 10^361. Here I was naively thinking that if no counter examples were found in the first, say, 10^10 numbers, no counter examples should exist. If only it…

John Conway showed that if you generalise the coefficients of the Collatz conjecture then some instances are undecidable. So maybe there is no proof.

Do you have a link?

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#40

Post by damiankennedy »

screenoridesaga wrote: post #14

I love how these raw mathematicians consider something proved when they can understand, meanwhile the computer can prove it easily just by counting a finite number of bits. What exactly would be considered proof in this case? Any explanation only mathematicians can understand?

The first thing you need to go with a series like this is prove that it converges. Then you can take a series that is already known to converge to pi. The choice of series will mean the difference between the proof being very hard and practically impossible. Then change that series to give 32/pi^3 instead of pi. Then deduct a mapping between groups of n in one series and groups of n in the other series.
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