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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

#41

Post by tsimionescu »

hyperhello wrote: post #36
feoren wrote: post #29

Earlier quoted context omitted.

> 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.

There are many series that converge to an extraordinarily small but non 0 number.

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

#42

Post by DavidSJ »

klysm wrote: post #17
messe wrote: post #9

Earlier quoted context omitted.

That's just the first counterexample that was proven to exist. The smallest counter example is less than 10^10 (in fact smaller than 10^9). > The Pólya conjecture was disproved by C. Brian Haselgrove in 1958. He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10^361.[3] > An explicit counterexample, of n = 906,180,359 was given by R. Sherman Lehman in 1960;[4] the smallest cou…

I believe inability to do arithmetic correctly is more common in folks with maths and other STEM degrees

It's one of the prerequisites in fact.

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

#43

Post by nickdrozd »

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…

> If it is undecidable, it is true.

That is the case for something like Goldbach's Conjecture, which says that every even number > 2 is the sum of two primes. If it's false, then there is a counterexample, and it is easy to prove whether or not a given number is a counterexample (just loop over all pairs of smaller primes).

But that is not the case for the Collatz Conjecture. A Collatz counterexample could be a number whose orbit loops back around. That would be a provable counterexample. Another kind of Collatz counterexample would be a number whose orbit never terminates or repeats, it just keeps going forever. If such an infinite sequence existed, it might not be possible to prove that it's infinite. And if it isn't provable, then the conjecture would both undecidable and false.

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

#44

Post by at_a_remove »

messe wrote: post #21
klysm wrote: post #17

Earlier quoted context omitted.

I believe inability to do arithmetic correctly is more common in folks with maths and other STEM degrees

I do like to joke that I studied maths because I don't like numbers.

I recall the end of a science fiction novel, wherein we finally meet aliens. It turns out that, as a trend, the species that are good at counting are bad at math and especially counting things at a glance. The species that is best at math have a hard time telling the difference between two things and three things just looking.

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

#45

Post by zeroonetwothree »

tzs wrote: post #38

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 ou…

What a weird coincidence, I was just looking at this paper today for a different reason.

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

#46

Post by gnatman »

>> One annoying thing about both the Collatz and Goldbach conjectures is that if they are unprovable, it's also impossible to prove they're unprovable (unless math is inconsistent). This has been proved!

That’s the most fun thing I’ve read all day!

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

#47

Post by hinkley »

messe wrote: post #21
klysm wrote: post #17

Earlier quoted context omitted.

I believe inability to do arithmetic correctly is more common in folks with maths and other STEM degrees

I do like to joke that I studied maths because I don't like numbers.

I know a young man with dyscalculia and we were hoping when he got to symbolic math that his problems would diminish.

That has not born fruit so far.

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

#48

Post by bo1024 »

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?

Nothing wrong with asking the question. There might not be any practical applications in those fields, and it could still be a worthwhile, interesting, fun question.

But in this case, I'd bet that developing the math needed to solve it would end up with practical consequences. I think of your question sort of like "what are the practical applications of teaching an AI to master Starcraft?" Even if playing Starcraft isn't practical, the existence of such an AI implies lots of practical consequences.

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

#49

Post by susam »

klysm wrote: post #17
messe wrote: post #9

Earlier quoted context omitted.

That's just the first counterexample that was proven to exist. The smallest counter example is less than 10^10 (in fact smaller than 10^9). > The Pólya conjecture was disproved by C. Brian Haselgrove in 1958. He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10^361.[3] > An explicit counterexample, of n = 906,180,359 was given by R. Sherman Lehman in 1960;[4] the smallest cou…

I believe inability to do arithmetic correctly is more common in folks with maths and other STEM degrees

> I believe inability to do arithmetic correctly is more common in folks with maths and other STEM degrees

Reminds me of a story from https://www.ams.org/notices/200410/fea-grothendieck-part2.pd...>. Quoting it below:

One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”

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

#50

Post by jl6 »

klysm wrote: post #17
messe wrote: post #9

Earlier quoted context omitted.

That's just the first counterexample that was proven to exist. The smallest counter example is less than 10^10 (in fact smaller than 10^9). > The Pólya conjecture was disproved by C. Brian Haselgrove in 1958. He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10^361.[3] > An explicit counterexample, of n = 906,180,359 was given by R. Sherman Lehman in 1960;[4] the smallest cou…

I believe inability to do arithmetic correctly is more common in folks with maths and other STEM degrees

“I’m a mathematician, not an arithmetician!”
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