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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

#21

Post by messe »

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

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

#22

Post by red_trumpet »

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?

Well, how does your proposed computer proof look like? A computer can easily calculate both sides of the equation up to say, float precision. But that's not a proof; it only tells you that both numbers are near each other!

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

#23

Post by permo-w »

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

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

#24

Post by pulisse »

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…

There's a great Mathoverflow thread collecting examples of this sort of phenomenon (called "eventual counterexamples" in the thread), together with some interesting analysis (see Joel Hamkins' answer): https://mathoverflow.net/questions/15444/examples-of-eventua...

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

#25

Post by macrolocal »

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?

Sorry for the cliché, but "the most exciting phrase to hear in science, the one that heralds the most discoveries, is not "Eureka!" (I found it!) but 'That's funny...'" --Asimov

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

#26

Post by timeagain »

SilasX wrote: post #15
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…

Okay but the Collatz conjecture is a little different in that there can't just be a one-off counterexample: it's a statement about a sequence. The counterexample would have to be either a cycle (that excludes 4/2/1), or a sequence of numbers that keep spiraling up indefinitely. And they've proven that any cycle would have to be very long[1]. Either way, it would mean trivially unlocking a sequence of numbers that hap…

Right. So there might be some set of numbers way out there that are like the “sporadic simple group(s)” of the collatz conjecture. Following the same rules as everyone else, you end up with something otherwise unclassifiable.

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

#27

Post by xanderlewis »

SOTGO wrote: post #5
nomemory wrote: post #2

It still boggling we haven't proved the sum of pi + e is irrational. Who cares anyway?

It's not that mind-boggling. In general we need to use specific properties of numbers to prove whether they are irrational, and pi+e has very few useful properties to work from. Even proving pi is irrational is not trivial and it is one of the transcendental numbers we know the most about.

> we need to use specific properties of numbers to prove whether they are irrational

Indeed, one could say that all we know about numbers are their properties. A number is just an existence assertion about an object fulfilling some property (possibly uniquely).

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

#28

Post by cubefox »

messe wrote: post #9
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…

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…

> He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10361

1.845 × 10361 = 19116.045

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

#29

Post by feoren »

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.

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

#30

Post by messe »

cubefox wrote: post #28
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…

> He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10361 1.845 × 10361 = 19116.045

Thanks, it should be 10^361. I've fixed it.
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