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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

#61

Post by 1propionyl »

susam wrote: post #49
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 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 c…

This is absolutely my personal experience. I am absolutely awful at arithmetic, but I think pretty competent at decently advanced mathematics.

It's like they say: the only numbers a mathematician needs are 0, 1, and 2 (and just because it's not 1).

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

#62

Post by NotYourLawyer »

datameta wrote: post #53
bo1024 wrote: post #48

Earlier quoted context omitted.

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 i…

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.

As one example, number theory was considered pure math with no real-world applications for a long time. But then cryptography became a thing…

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

#63

Post by tristenharr »

ajkjk wrote: post #58
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…

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 treate…

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/1CERdyZMlGj76EmMJXt5Zr3pM...

https://docs.google.com/document/d/1LpKqgCzjo8LrxB547WY5YdVP...

https://docs.google.com/document/d/1JvV5NJTYaTHn3eSAShcUtE8J...

https://docs.google.com/document/d/1Ej0bNIpYNT30M31pCxx-LDoe...

https://docs.google.com/document/d/13nlw3vbxyRGd6pYeAre8c7e5...

https://docs.google.com/document/d/1WChBcGtaUdLckPDqNBVGpP9x...

https://docs.google.com/document/d/1_bQbgJ5HIyv_D5ex9TWK2XxH...

https://docs.google.com/document/d/1AomtTAP4GBXpOuArvBYQsHiP...

https://docs.google.com/document/d/1QO8iQ2OmPZkQ7lNS0uUB0uyy...

https://docs.google.com/document/d/1zzedxz1Ya9rT8N4RZZNtpsrr...

https://docs.google.com/document/d/1vB0Qbqe0yiO0tcxswl2jniYc...

https://docs.google.com/document/d/1K4upBz2awHnKlr5JE1xocntp...

https://docs.google.com/document/d/1XhOIy0RO59CrZEXXPK-QAR_s...

https://docs.google.com/document/d/1rJx2KHQ0GqZfESqyFzXGLmN_...

https://docs.google.com/document/d/19kZrg8HvxZ31-goiwL0XLyPC...

https://docs.google.com/document/d/1F8c_gVy47bRw5zZVWKb7x5SV...

https://docs.google.com/document/d/16JolkQcHD3eviZuakfT7iam7...

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

#64

Post by eru »

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…

> EDIT: rephrased because despite having a maths degree I can't count.

You would probably enjoy Tom Körner's book 'The Pleasures of Counting'. See eg https://maa.org/press/maa-reviews/the-pleasures-of-counting

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

#65

Post by eru »

1propionyl wrote: post #61
susam wrote: post #49

Earlier quoted context omitted.

> 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 c…

This is absolutely my personal experience. I am absolutely awful at arithmetic, but I think pretty competent at decently advanced mathematics. It's like they say: the only numbers a mathematician needs are 0, 1, and 2 (and just because it's not 1).

In number theory, 2 is often a special case in a lot of theorems. There's something odd about the even prime.

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

#66

Post by eru »

NotYourLawyer wrote: post #62
datameta wrote: post #53

Earlier quoted context omitted.

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.

As one example, number theory was considered pure math with no real-world applications for a long time. But then cryptography became a thing…

Cryptography and coding theory.

Cryptography and coding theory and compression are three sides of the same (generalised) coin. Claude Shannon dabbled in all of them.

See eg https://www.dpmms.cam.ac.uk/~twk/Shan.pdf

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

#67

Post by anthomtb »

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 pursued and achieved an EE degree plus a couple extra math courses largely because I was "not good at math" as a youth. Nothing like hearing "I don't think math is your subject" from a key adult to light a fire under ones fanny.

In hindsight I could have had the same career path with a straight CS degree and much less stress during my college years. No regrets though, math really is fun! Even more so when the classroom setting is removed.

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

#68

Post by passion__desire »

tristenharr wrote: post #63
ajkjk wrote: post #58

Earlier quoted context omitted.

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 treate…

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 ha…

Thanks for being so candid. All I have to add is if you think a particular idea is driving you crazy or could be your next breakthrough, just let that idea simmer for sometime. You will get clarity over time or you will drop the idea completely. This strategy has worked out quite well for me.

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

#69

Post by Paul-Craft »

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…

One of my favorite mathematical tangents involves a theorem of Littlewood.

The prime number theorem states that the prime counting function $\pi(x)$ is well approximated by the integral $\int_0^x \frac{dt} {\log t},$ which is a function that's become named $\mathrm{li}(x).$ Littlewood proved that the sign of $\pi(x) - \li(x)$ changes infinitely often, but his proof didn't produce a specific value where such a sign change occurs.

In 1933, one of Littlewood's students showed that, assuming the Riemann hypothesis, at least one sign change occurs below the number $e^e^e^79$, which is approximately $10^10^10^34.$ In 1955, he was able to show unconditionally that such a sign change occurs below the number $e^e^e^e^7.705$, which is approximately $10^10^10^964$.

Both of those numbers absolutely dwarf the estimated number of elementary particles in the observable universe, so it's utterly impossible to either compute or write them down, even if you used the entire universe as your computer or your writing surface.

https://en.wikipedia.org/wiki/Skewes%27s_number

That upper bound has since been lowered in 1999 to 1.38922 * 10^316 here: https://www.ams.org/journals/mcom/2000-69-231/S0025-5718-99-...

I imagine it there might have been a bit more progress in the last 25 years, but I have no idea if it's possible to actually write down all the digits of the current best upper bound.

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

#70

Post by bowsamic »

tristenharr wrote: post #63
ajkjk wrote: post #58

Earlier quoted context omitted.

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 treate…

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 ha…

Those documents are a bit terrifying, especially when the bold and capital letters come out
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