[HN Gopher] The seventh most popular easily understood unsolved ...
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The seventh most popular easily understood unsolved problem on
MathOverflow
Author : ColinWright
Score : 61 points
Date : 2024-02-16 19:01 UTC (3 hours ago)
(HTM) web link (mathstodon.xyz)
(TXT) w3m dump (mathstodon.xyz)
| nomemory wrote:
| It still boggling we haven't proved the sum of pi + e is
| irrational. Who cares anyway?
| xanderlewis wrote:
| > Who cares anyway?
|
| I can't tell if you're just joking. Proofs of facts that seem
| obvious or irrelevant are very important not for the yes-no
| question they answer but for the development of the deep
| mathematical theory they necessitate.
| nomemory wrote:
| It was a joke, I know there are lots of people who care about
| that.
| SOTGO wrote:
| 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.
| xanderlewis wrote:
| > 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).
| nerdponx wrote:
| > fairly simple
|
| _That 's_ your definition of a fairly simple identity?!
| nomemory wrote:
| In his defense lots of unsolved questions are inscrutable for
| the majority of people. So this one looks decent.
| Almondsetat wrote:
| it fits in a line. it's simple
| ajkjk wrote:
| Well it's something an undergraduate can understand and in
| principle compute/play with. That's pretty simple,
| comparatively.
| Buttons840 wrote:
| 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 were so easy.
|
| [0]: https://en.wikipedia.org/wiki/P%C3%B3lya_conjecture
| messe wrote:
| 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 Polya 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 x 10^361.[3]
|
| > An explicit counterexample, of n = 906,180,359 was given by
| R. Sherman Lehman in 1960;[4] the smallest counterexample is n
| = 906,150,257, found by Minoru Tanaka in 1980.[5]
|
| EDIT: rephrased because despite having a maths degree I can't
| count.
| klysm wrote:
| I believe inability to do arithmetic correctly is more common
| in folks with maths and other STEM degrees
| messe wrote:
| I do like to joke that I studied maths because I don't like
| numbers.
| at_a_remove wrote:
| [delayed]
| DavidSJ wrote:
| It's one of the prerequisites in fact.
| cubefox wrote:
| > He showed that the conjecture has a counterexample, which
| he estimated to be around 1.845 x 10361
|
| 1.845 x 10361 = 19116.045
| messe wrote:
| Thanks, it should be 10^361. I've fixed it.
| senderista wrote:
| See also https://mathoverflow.net/questions/95865/examples-of-
| conject...
| Shawnecy wrote:
| That reminds me of a parody song from 3Blue1Brown[0] about how
| certain numerical patterns don't quite hold and can fool you if
| you don't look far enough out.
|
| [0]: https://www.youtube.com/watch?v=NOCsdhzo6Jg
| SilasX wrote:
| 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 happens to
| satisfy a very specific criterion.
|
| [1]
| https://en.wikipedia.org/wiki/Collatz_conjecture?useskin=vec...
| timeagain wrote:
| 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.
| eutectic wrote:
| John Conway showed that if you generalise the coefficients of
| the Collatz conjecture then some instances are undecidable. So
| maybe there is no proof.
| cubefox wrote:
| 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.
| dooglius wrote:
| 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
| nickdrozd wrote:
| > 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.
| thesz wrote:
| Do you have a link?
| permo-w wrote:
| >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
| zimpenfish wrote:
| > naively
|
| I think that answers your question?
| pulisse wrote:
| 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...
| coolThingsFirst wrote:
| Im in the middle of something but that doesn't look that hard
| paulpauper wrote:
| Gourevitch's conjecture is based on a transformed Ramanujan pi
| formula via Dougall's identity .
|
| more info
|
| https://www.ams.org/journals/mcom/2014-83-285/S0025-5718-201...
| screenoridesaga wrote:
| 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?
| earlymodernlvr wrote:
| The computer can only check a finite number of cases
| computationally, a proof can prove the results for all N.
| red_trumpet wrote:
| 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!
| feoren wrote:
| > 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.
| hyperhello wrote:
| 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.
| tsimionescu wrote:
| There are many series that converge to an extraordinarily
| small but non 0 number.
| damiankennedy wrote:
| 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.
| datameta wrote:
| 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?
| macrolocal wrote:
| Sorry for the cliche, 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
| Sharlin wrote:
| None that we know of. We still do it because it's worth doing.
| tzs wrote:
| 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) < H(n) +
| exp(H(n)) log(H(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
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