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The seventh most popular easily understood unsolved problem on MathOverflow

187 points by ColinWright 3y ago

nomemory 3y ago
It still boggling we haven't proved the sum of pi + e is irrational. Who cares anyway?
- xanderlewis 3y ago
  > 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 3y ago
    It was a joke, I know there are lots of people who care about that.
- SOTGO 3y ago
  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 3y ago
    > 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 3y ago
> fairly simple That's your definition of a fairly simple identity?!
- nomemory 3y ago
  In his defense lots of unsolved questions are inscrutable for the majority of people. So this one looks decent.
- Almondsetat 3y ago
  it fits in a line. it's simple
- ajkjk 3y ago
  Well it's something an undergraduate can understand and in principle compute/play with. That's pretty simple, comparatively.
Buttons840 3y ago
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 https://en.wikipedia.org/wiki/P%C3%B3lya_conjecture
- messe 3y ago
  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 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 3y ago
    I believe inability to do arithmetic correctly is more common in folks with maths and other STEM degrees
    - messe 3y ago
      I do like to joke that I studied maths because I don't like numbers.
      - at_a_remove 3y ago
        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.
coolThingsFirst 3y ago
Im in the middle of something but that doesn’t look that hard
- Alifatisk 3y ago
  Why did I find this funny?
  - coolThingsFirst 3y ago
    Because you aren't dead inside and can take a joke.
paulpauper 3y ago
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/S0025-5718-2013-02701-9/S0025-5718-2013-02701-9.pdf https://www.ams.org/journals/mcom/2014-83-285/S0025-5718-201...
screenoridesaga 3y ago
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 3y ago
  The computer can only check a finite number of cases computationally, a proof can prove the results for all N.
- red_trumpet 3y ago
  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 3y ago
  > 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 3y ago
    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 3y ago
      There are many series that converge to an extraordinarily small but non 0 number.
- damiankennedy 3y ago
  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 3y ago
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 3y ago
  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
- Sharlin 3y ago
  None that we know of. We still do it because it's worth doing.
  - datameta 3y ago
    I do believe it is worth doing in and of itself. So my question is not trying to get a gotcha if not supplied with a practical application. I think I could certainly improve my intuition in theoretical math fields.
- bo1024 3y ago
  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.
  - datameta 3y ago
    Ah I see. As we evolve our understanding and toolset of certain fields in math, we may serendipitously happen upon approaches to solve existing problems or solve future ones, perhaps.
tzs 3y ago
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 https://arxiv.org/abs/math/0008177
- zeroonetwothree 3y ago
  What a weird coincidence, I was just looking at this paper today for a different reason.
  - renonce 3y ago
    Care to tell us the reason?
- ajkjk 3y ago
  I've always felt that the Riemann hypothesis' specific formulation, in terms of the zeroes of zeta on the critical strip, is a very annoying way of expressing it. Like it is clearly related to the prime numbers in a really intrinsic way and therefore the most "intrinsic" expression of the problem shouldn't have anything to do with zeta or complex numbers or roots at all... whatever that is, I wish it's what we treated as the big unsolved problem instead of what we got. I wonder what the most pleasing equivalent formulation of the RH would be. Gotta assume that some of them are much more directly about prime numbers. Yours seems pretty good. There are some others on here: https://mathoverflow.net/questions/39944/collection-of-equivalent-forms-of-riemann-hypothesis https://mathoverflow.net/questions/39944/collection-of-equiv... which seem good also.
  - tristenharr 3y ago
    I’m bipolar and when I was first diagnosed with bipolar disorder I was 19 and in my first ever manic episode I dropped out of college and went deep down this particular rabbit-hole in an attempt to solve the Riemann Hypothesis to win the prize. I spent 3 months on it gradually driving myself crazier and crazier. My second company had collapsed out from under me when a sociopathic employee with a gambling addiction had conned me into thinking they were my best friend and meanwhile skimmed all of our profits so it looked like we were barely breaking even right under my nose. (To be fair I was 19) That in combination with a semester of Intro to Philosophy where I read Aristotle and Socrates and a Algebra class I became obsessed with led to me stumbling into the Riemann Hypothesis, and going deep down a rabbit hole in what ultimately turned into my first ever and most intense manic episode and led to my bipolar diagnosis. I hadn’t even finished college Algebra at the time. I basically spent months playing around with equations on Wolfram Alpha. The funny thing is this ultimately was kind of a life-changing moment for me, as I learned programming through it which in combination with a family member dying is what finally pulled me out of the mania, and then I changed my major to Computer Science re-enrolled in college and have since actually gone on to get a degree in CS and a minor in math, sometimes I look back at the giant set of Google Docs I created during that time, mostly in amusement but to this day some things I look at and don’t even understand if it is meaningless or not. A part of me wants to pick it back up, but then I tell myself it isn’t worth the risk, I’ve always felt I am uniquely qualified to understand how the guy in the movie “A beautiful mind” felt when he stopped talking/engaging with the hallucinations. Here’s some links to random Google docs from my ~2017 rabbit hole if anyone is interested. There’s a lot of fairly interesting usage of the Euler-Mascheroni constant and honestly it goes a bit all over the place. I kind of doubt there’s anything here, but if you’ve ever wanted to peek into unchecked mania.. have fun. It was months of insanity. (Sharing these in good faith!) https://docs.google.com/document/d/1CERdyZMlGj76EmMJXt5Zr3pMcBeTeI7uFmIp0aRmX2M/edit https://docs.google.com/document/d/1CERdyZMlGj76EmMJXt5Zr3pM... https://docs.google.com/document/d/1LpKqgCzjo8LrxB547WY5YdVPVsSm6mpw4RT6WGunhGs/edit https://docs.google.com/document/d/1LpKqgCzjo8LrxB547WY5YdVP... https://docs.google.com/document/d/1JvV5NJTYaTHn3eSAShcUtE8Jnn95Eof7raNtRfO87QQ/edit https://docs.google.com/document/d/1JvV5NJTYaTHn3eSAShcUtE8J... https://docs.google.com/document/d/1Ej0bNIpYNT30M31pCxx-LDoe5x3ZJ5OnwP7APcvBT64/edit https://docs.google.com/document/d/1Ej0bNIpYNT30M31pCxx-LDoe... https://docs.google.com/document/d/13nlw3vbxyRGd6pYeAre8c7e5OKmPjjksGaKL422pFfE/edit https://docs.google.com/document/d/13nlw3vbxyRGd6pYeAre8c7e5... https://docs.google.com/document/d/1WChBcGtaUdLckPDqNBVGpP9xaYqpP02GflFMyWbl_IQ/edit https://docs.google.com/document/d/1WChBcGtaUdLckPDqNBVGpP9x... https://docs.google.com/document/d/1_bQbgJ5HIyv_D5ex9TWK2XxH1RDBZx3tTIoZMJeX8zk/edit https://docs.google.com/document/d/1_bQbgJ5HIyv_D5ex9TWK2XxH... https://docs.google.com/document/d/1AomtTAP4GBXpOuArvBYQsHiPRfyAzWZGC-E37v5-Fhw/edit https://docs.google.com/document/d/1AomtTAP4GBXpOuArvBYQsHiP... https://docs.google.com/document/d/1QO8iQ2OmPZkQ7lNS0uUB0uyyWAf2NID85fqfTP9a7Fs/edit https://docs.google.com/document/d/1QO8iQ2OmPZkQ7lNS0uUB0uyy... https://docs.google.com/document/d/1zzedxz1Ya9rT8N4RZZNtpsrrNkLLPfEtOqZBqapNNR0/edit https://docs.google.com/document/d/1zzedxz1Ya9rT8N4RZZNtpsrr... https://docs.google.com/document/d/1vB0Qbqe0yiO0tcxswl2jniYcXjfzTewWEMfXXU0J8rw/edit https://docs.google.com/document/d/1vB0Qbqe0yiO0tcxswl2jniYc... https://docs.google.com/document/d/1K4upBz2awHnKlr5JE1xocntpQCuu7IWJW7iuevJTJXk/edit https://docs.google.com/document/d/1K4upBz2awHnKlr5JE1xocntp... https://docs.google.com/document/d/1XhOIy0RO59CrZEXXPK-QAR_smBhRPewq0LINUw6pAU0/edit https://docs.google.com/document/d/1XhOIy0RO59CrZEXXPK-QAR_s... https://docs.google.com/document/d/1rJx2KHQ0GqZfESqyFzXGLmN_5H0iRxIhJbLJLr9fj5M/edit https://docs.google.com/document/d/1rJx2KHQ0GqZfESqyFzXGLmN_... https://docs.google.com/document/d/19kZrg8HvxZ31-goiwL0XLyPCh3XPVCXaBDmW_67-fHw/edit https://docs.google.com/document/d/19kZrg8HvxZ31-goiwL0XLyPC... https://docs.google.com/document/d/1F8c_gVy47bRw5zZVWKb7x5SVR0vruNk2XbTnOH5vAz0/edit https://docs.google.com/document/d/1F8c_gVy47bRw5zZVWKb7x5SV... https://docs.google.com/document/d/16JolkQcHD3eviZuakfT7iam7zCot3QPoVkD-2jbg7Xs/edit https://docs.google.com/document/d/16JolkQcHD3eviZuakfT7iam7...
gnatman 3y ago
>> 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!
- dtgriscom 3y ago
  Prove it.
- Mimmy 3y ago
  one of the saddest too
daxfohl 3y ago
> 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! So is there a countable order of metaprovability? i.e. are there problems that if they're impossible to prove, it's impossible to prove whether it's possible to prove whether they're unprovable, but _that_ can be proven? Or does that mean the same thing? What does it mean if that goes infinitely deep? That we'll never know anything about a problem in that set unless we prove it? I guess it's not possible to prove that a problem is in that set, right? Is it possible to prove that problems do exist in that set at all? edit: (non-vacuously. Provable problems are obviously vacuously in all the sets). edit2: oh I see now (I think). Above I'd need an "AND if you can't prove it undecideable ... then you can't prove its undecidability's decideability either, and that can be proved" So basically each level needs to be conditioned on all 0..n-2 levels not holding, then you can prove level n-1 is undecideable. level 0: provable level 1: provably undecideable level 2: if not provable (can't prove level 0) then can't prove undecideable. (Goldbach, Collatz) level 3: if it's not in level 0 or 1, then can't prove level 2. level 4: if it's not in levels 0, 1, or 2, then can't prove level 3. ... Which means there's no level infinity, since there's no level infinity minus 1 to talk about so it's meaningless. Correct?
- deleted 3y ago
  [deleted]
- hansvm 3y ago
  A couple of things: > level 2: if not provable (can't prove level 0) then can't prove undecideable. (Goldbach, Collatz) If you're operating in the ordinary naturals, neither Goldbach nor Collatz are undecidable. The statements are either true or false. The question at hand is whether we can generate a finite proof of that fact in a given axiomatic system. The quote at hand simply said that if no such proof exists (obviously implying the statements are true because otherwise a simple finite proof would be a counter-example) then similarly no finite proof of that lack of existence exists either. They might be undecidable in other arithmetic systems (implying that some models of those systems would have the statements be true and some would have them be false), but not for the naturals. > Which means there's no level infinity, since there's no level infinity minus 1 to talk about so it's meaningless. Correct? When you're making up a new definition, you care about (1) is it coherent (don't want to be like the proverbial Ph.D. who made up an exciting mathematical object and studied it for years before finding out no such objects could exist), (2) what can you deduce about that object, and (3) is it "useful" (for some broad definition thereof). The question of "meaningfulness" can lead you down incorrect paths because it merges (2) and (3). The definition you picked for the broad concept you have in your mind has no level "infinity" because you snuck the naturals into the definition as the indexing set (defined using the successor function). If we strictly look at point (2), yes, there's no "infinity". The general concept you're looking at though might more naturally fit in a world where there are infinities (i.e., let's look at point (3) a bit more). As a rule of thumb, if you're looking at a set of things indexed by the naturals, especially if they have a successor function, especially if they're defined inductively, especially if their interesting properties are defined in terms of all previous items, it's natural to take a look at indexing via the ordinals and trying to define them via transfinite induction. For this particular set of things, that may or may not be straightforward. The notions of unprovability, undecidability, ... are a bit intricate, and you need to define them with respect to the system being studied and the system being used to study them. Right now, your inductive definition _seems_ to also require a predecessor function (which, if intrinsic to the property being analyzed would preclude many attempts to wrangle in some infinities), but I wouldn't be surprised if a careful re-writing found that to be an extraneous detail, in which case you could add in a limiting case (defining f(w) in terms of f(n) for all n<w for all limit ordinals w).
EVa5I7bHFq9mnYK 3y ago
I'm sorry, but who needs this? If the answer is "we will discover new math methods trying to solve that puzzle", isn't it better to discover new math methods trying to solve something useful?
- medstrom 3y ago
  The history of math is rife with useless puzzles generating useful solutions for other things.
- wruza 3y ago
  How can you see something useful if there’s no way to even think about it?
Paul-Craft 3y ago
Just out of curiosity, did anybody chase down the paper referenced in the post? Here it is: https://www.emis.de/journals/EM/expmath/volumes/12/12.4/Guillera.pdf https://www.emis.de/journals/EM/expmath/volumes/12/12.4/Guil... The author of the paper actually notes that a slightly different series is suspected to converge to $32/\pi^3$. The good news is that that series is an alternating series, so it's guaranteed to converge to something. But, it converges so slowly that, although the second partial sum differs from $32/pi^3$ by about 0.0004328909249, the 5000th partial sum differs by about 0.0004330860787. Those two numbers only start to differ at the 6th digit beyond the decimal point.
tocs3 3y ago
No Wikipedia article?

187 points by ColinWright 3y ago

nomemory 3y ago

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

xanderlewis 3y ago
> 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 3y ago
  It was a joke, I know there are lots of people who care about that.
SOTGO 3y ago
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 3y ago
  > 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 3y ago

> fairly simple That's your definition of a fairly simple identity?!

nomemory 3y ago
In his defense lots of unsolved questions are inscrutable for the majority of people. So this one looks decent.
Almondsetat 3y ago
it fits in a line. it's simple
ajkjk 3y ago
Well it's something an undergraduate can understand and in principle compute/play with. That's pretty simple, comparatively.

Buttons840 3y ago

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 https://en.wikipedia.org/wiki/P%C3%B3lya_conjecture

messe 3y ago
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 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 3y ago
  I believe inability to do arithmetic correctly is more common in folks with maths and other STEM degrees
  - messe 3y ago
    I do like to joke that I studied maths because I don't like numbers.
    - at_a_remove 3y ago
      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.

coolThingsFirst 3y ago

Im in the middle of something but that doesn’t look that hard

Alifatisk 3y ago
Why did I find this funny?
- coolThingsFirst 3y ago
  Because you aren't dead inside and can take a joke.

paulpauper 3y ago

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/S0025-5718-2013-02701-9/S0025-5718-2013-02701-9.pdf https://www.ams.org/journals/mcom/2014-83-285/S0025-5718-201...

screenoridesaga 3y ago

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 3y ago
The computer can only check a finite number of cases computationally, a proof can prove the results for all N.
red_trumpet 3y ago
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 3y ago
> 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 3y ago
  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 3y ago
    There are many series that converge to an extraordinarily small but non 0 number.
damiankennedy 3y ago
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 3y ago

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 3y ago
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
Sharlin 3y ago
None that we know of. We still do it because it's worth doing.
- datameta 3y ago
  I do believe it is worth doing in and of itself. So my question is not trying to get a gotcha if not supplied with a practical application. I think I could certainly improve my intuition in theoretical math fields.
bo1024 3y ago
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.
- datameta 3y ago
  Ah I see. As we evolve our understanding and toolset of certain fields in math, we may serendipitously happen upon approaches to solve existing problems or solve future ones, perhaps.

tzs 3y ago

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 https://arxiv.org/abs/math/0008177

zeroonetwothree 3y ago
What a weird coincidence, I was just looking at this paper today for a different reason.
- renonce 3y ago
  Care to tell us the reason?
ajkjk 3y ago
I've always felt that the Riemann hypothesis' specific formulation, in terms of the zeroes of zeta on the critical strip, is a very annoying way of expressing it. Like it is clearly related to the prime numbers in a really intrinsic way and therefore the most "intrinsic" expression of the problem shouldn't have anything to do with zeta or complex numbers or roots at all... whatever that is, I wish it's what we treated as the big unsolved problem instead of what we got. I wonder what the most pleasing equivalent formulation of the RH would be. Gotta assume that some of them are much more directly about prime numbers. Yours seems pretty good. There are some others on here: https://mathoverflow.net/questions/39944/collection-of-equivalent-forms-of-riemann-hypothesis https://mathoverflow.net/questions/39944/collection-of-equiv... which seem good also.
- tristenharr 3y ago
  I’m bipolar and when I was first diagnosed with bipolar disorder I was 19 and in my first ever manic episode I dropped out of college and went deep down this particular rabbit-hole in an attempt to solve the Riemann Hypothesis to win the prize. I spent 3 months on it gradually driving myself crazier and crazier. My second company had collapsed out from under me when a sociopathic employee with a gambling addiction had conned me into thinking they were my best friend and meanwhile skimmed all of our profits so it looked like we were barely breaking even right under my nose. (To be fair I was 19) That in combination with a semester of Intro to Philosophy where I read Aristotle and Socrates and a Algebra class I became obsessed with led to me stumbling into the Riemann Hypothesis, and going deep down a rabbit hole in what ultimately turned into my first ever and most intense manic episode and led to my bipolar diagnosis. I hadn’t even finished college Algebra at the time. I basically spent months playing around with equations on Wolfram Alpha. The funny thing is this ultimately was kind of a life-changing moment for me, as I learned programming through it which in combination with a family member dying is what finally pulled me out of the mania, and then I changed my major to Computer Science re-enrolled in college and have since actually gone on to get a degree in CS and a minor in math, sometimes I look back at the giant set of Google Docs I created during that time, mostly in amusement but to this day some things I look at and don’t even understand if it is meaningless or not. A part of me wants to pick it back up, but then I tell myself it isn’t worth the risk, I’ve always felt I am uniquely qualified to understand how the guy in the movie “A beautiful mind” felt when he stopped talking/engaging with the hallucinations. Here’s some links to random Google docs from my ~2017 rabbit hole if anyone is interested. There’s a lot of fairly interesting usage of the Euler-Mascheroni constant and honestly it goes a bit all over the place. I kind of doubt there’s anything here, but if you’ve ever wanted to peek into unchecked mania.. have fun. It was months of insanity. (Sharing these in good faith!) https://docs.google.com/document/d/1CERdyZMlGj76EmMJXt5Zr3pMcBeTeI7uFmIp0aRmX2M/edit https://docs.google.com/document/d/1CERdyZMlGj76EmMJXt5Zr3pM... https://docs.google.com/document/d/1LpKqgCzjo8LrxB547WY5YdVPVsSm6mpw4RT6WGunhGs/edit https://docs.google.com/document/d/1LpKqgCzjo8LrxB547WY5YdVP... https://docs.google.com/document/d/1JvV5NJTYaTHn3eSAShcUtE8Jnn95Eof7raNtRfO87QQ/edit https://docs.google.com/document/d/1JvV5NJTYaTHn3eSAShcUtE8J... https://docs.google.com/document/d/1Ej0bNIpYNT30M31pCxx-LDoe5x3ZJ5OnwP7APcvBT64/edit https://docs.google.com/document/d/1Ej0bNIpYNT30M31pCxx-LDoe... https://docs.google.com/document/d/13nlw3vbxyRGd6pYeAre8c7e5OKmPjjksGaKL422pFfE/edit https://docs.google.com/document/d/13nlw3vbxyRGd6pYeAre8c7e5... https://docs.google.com/document/d/1WChBcGtaUdLckPDqNBVGpP9xaYqpP02GflFMyWbl_IQ/edit https://docs.google.com/document/d/1WChBcGtaUdLckPDqNBVGpP9x... https://docs.google.com/document/d/1_bQbgJ5HIyv_D5ex9TWK2XxH1RDBZx3tTIoZMJeX8zk/edit https://docs.google.com/document/d/1_bQbgJ5HIyv_D5ex9TWK2XxH... https://docs.google.com/document/d/1AomtTAP4GBXpOuArvBYQsHiPRfyAzWZGC-E37v5-Fhw/edit https://docs.google.com/document/d/1AomtTAP4GBXpOuArvBYQsHiP... https://docs.google.com/document/d/1QO8iQ2OmPZkQ7lNS0uUB0uyyWAf2NID85fqfTP9a7Fs/edit https://docs.google.com/document/d/1QO8iQ2OmPZkQ7lNS0uUB0uyy... https://docs.google.com/document/d/1zzedxz1Ya9rT8N4RZZNtpsrrNkLLPfEtOqZBqapNNR0/edit https://docs.google.com/document/d/1zzedxz1Ya9rT8N4RZZNtpsrr... https://docs.google.com/document/d/1vB0Qbqe0yiO0tcxswl2jniYcXjfzTewWEMfXXU0J8rw/edit https://docs.google.com/document/d/1vB0Qbqe0yiO0tcxswl2jniYc... https://docs.google.com/document/d/1K4upBz2awHnKlr5JE1xocntpQCuu7IWJW7iuevJTJXk/edit https://docs.google.com/document/d/1K4upBz2awHnKlr5JE1xocntp... https://docs.google.com/document/d/1XhOIy0RO59CrZEXXPK-QAR_smBhRPewq0LINUw6pAU0/edit https://docs.google.com/document/d/1XhOIy0RO59CrZEXXPK-QAR_s... https://docs.google.com/document/d/1rJx2KHQ0GqZfESqyFzXGLmN_5H0iRxIhJbLJLr9fj5M/edit https://docs.google.com/document/d/1rJx2KHQ0GqZfESqyFzXGLmN_... https://docs.google.com/document/d/19kZrg8HvxZ31-goiwL0XLyPCh3XPVCXaBDmW_67-fHw/edit https://docs.google.com/document/d/19kZrg8HvxZ31-goiwL0XLyPC... https://docs.google.com/document/d/1F8c_gVy47bRw5zZVWKb7x5SVR0vruNk2XbTnOH5vAz0/edit https://docs.google.com/document/d/1F8c_gVy47bRw5zZVWKb7x5SV... https://docs.google.com/document/d/16JolkQcHD3eviZuakfT7iam7zCot3QPoVkD-2jbg7Xs/edit https://docs.google.com/document/d/16JolkQcHD3eviZuakfT7iam7...

gnatman 3y ago

>> 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!

dtgriscom 3y ago
Prove it.
Mimmy 3y ago
one of the saddest too

daxfohl 3y ago

> 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! So is there a countable order of metaprovability? i.e. are there problems that if they're impossible to prove, it's impossible to prove whether it's possible to prove whether they're unprovable, but _that_ can be proven? Or does that mean the same thing? What does it mean if that goes infinitely deep? That we'll never know anything about a problem in that set unless we prove it? I guess it's not possible to prove that a problem is in that set, right? Is it possible to prove that problems do exist in that set at all? edit: (non-vacuously. Provable problems are obviously vacuously in all the sets). edit2: oh I see now (I think). Above I'd need an "AND if you can't prove it undecideable ... then you can't prove its undecidability's decideability either, and that can be proved" So basically each level needs to be conditioned on all 0..n-2 levels not holding, then you can prove level n-1 is undecideable. level 0: provable level 1: provably undecideable level 2: if not provable (can't prove level 0) then can't prove undecideable. (Goldbach, Collatz) level 3: if it's not in level 0 or 1, then can't prove level 2. level 4: if it's not in levels 0, 1, or 2, then can't prove level 3. ... Which means there's no level infinity, since there's no level infinity minus 1 to talk about so it's meaningless. Correct?

deleted 3y ago
[deleted]
hansvm 3y ago
A couple of things: > level 2: if not provable (can't prove level 0) then can't prove undecideable. (Goldbach, Collatz) If you're operating in the ordinary naturals, neither Goldbach nor Collatz are undecidable. The statements are either true or false. The question at hand is whether we can generate a finite proof of that fact in a given axiomatic system. The quote at hand simply said that if no such proof exists (obviously implying the statements are true because otherwise a simple finite proof would be a counter-example) then similarly no finite proof of that lack of existence exists either. They might be undecidable in other arithmetic systems (implying that some models of those systems would have the statements be true and some would have them be false), but not for the naturals. > Which means there's no level infinity, since there's no level infinity minus 1 to talk about so it's meaningless. Correct? When you're making up a new definition, you care about (1) is it coherent (don't want to be like the proverbial Ph.D. who made up an exciting mathematical object and studied it for years before finding out no such objects could exist), (2) what can you deduce about that object, and (3) is it "useful" (for some broad definition thereof). The question of "meaningfulness" can lead you down incorrect paths because it merges (2) and (3). The definition you picked for the broad concept you have in your mind has no level "infinity" because you snuck the naturals into the definition as the indexing set (defined using the successor function). If we strictly look at point (2), yes, there's no "infinity". The general concept you're looking at though might more naturally fit in a world where there are infinities (i.e., let's look at point (3) a bit more). As a rule of thumb, if you're looking at a set of things indexed by the naturals, especially if they have a successor function, especially if they're defined inductively, especially if their interesting properties are defined in terms of all previous items, it's natural to take a look at indexing via the ordinals and trying to define them via transfinite induction. For this particular set of things, that may or may not be straightforward. The notions of unprovability, undecidability, ... are a bit intricate, and you need to define them with respect to the system being studied and the system being used to study them. Right now, your inductive definition _seems_ to also require a predecessor function (which, if intrinsic to the property being analyzed would preclude many attempts to wrangle in some infinities), but I wouldn't be surprised if a careful re-writing found that to be an extraneous detail, in which case you could add in a limiting case (defining f(w) in terms of f(n) for all n<w for all limit ordinals w).

EVa5I7bHFq9mnYK 3y ago

I'm sorry, but who needs this? If the answer is "we will discover new math methods trying to solve that puzzle", isn't it better to discover new math methods trying to solve something useful?

medstrom 3y ago
The history of math is rife with useless puzzles generating useful solutions for other things.
wruza 3y ago
How can you see something useful if there’s no way to even think about it?

Paul-Craft 3y ago

Just out of curiosity, did anybody chase down the paper referenced in the post? Here it is: https://www.emis.de/journals/EM/expmath/volumes/12/12.4/Guillera.pdf https://www.emis.de/journals/EM/expmath/volumes/12/12.4/Guil... The author of the paper actually notes that a slightly different series is suspected to converge to $32/\pi^3$. The good news is that that series is an alternating series, so it's guaranteed to converge to something. But, it converges so slowly that, although the second partial sum differs from $32/pi^3$ by about 0.0004328909249, the 5000th partial sum differs by about 0.0004330860787. Those two numbers only start to differ at the 6th digit beyond the decimal point.

tocs3 3y ago

No Wikipedia article?