5 ms·

Could someone elaborate on why this is interesting? The linked page doesn't have much context.

JshWright 7y ago
Numberphile is a great resource for pretty much any "pop math" topic. https://www.youtube.com/watch?v=ASoz_NuIvP0 https://www.youtube.com/watch?v=ASoz_NuIvP0
- ArneVogel 7y ago
  They also just released a video on 42: https://www.youtube.com/watch?v=zyG8Vlw5aAw https://www.youtube.com/watch?v=zyG8Vlw5aAw
ajross 7y ago
The form of the statement sounds like something related to Fermat's last theorem, I guess?
fl0wenol 7y ago
It is an open question as to whether every integer not equal to 4 or 5 modulo 9 is the sum of three cubes, although it is suspected to be true. https://en.wikipedia.org/wiki/Sums_of_three_cubes#Computational_results https://en.wikipedia.org/wiki/Sums_of_three_cubes#Computatio... 33 and 42 were known to be exceptions of all sums less than 100 for which solutions were found, until recently. 42 was the most recent to fall.
- huntermonk 7y ago
  Thanks, this is helpful. Why do we care if every integer not equal to 4 or 5 modulo 9 is the sum of three cubes? Just for fun?
  - ipince 7y ago
    Well, yeah, pretty much. In number theory you study properties/patterns of the numbers. This is an example of such a property. These properties often seem like they don’t have any applications, and often they don’t, but not always. For example, cryptography is basically all based on number theory.
    - seph-reed 7y ago
      I watched a great series once on generating ~prime numbers which explained how to use number theory to check a randomly generated number for a probability of being prime. By using a bunch of weird rules, you can be ~99% sure it's prime without having to do huge computations.
  - tejtm 7y ago
    Some folks are attracted to ways of thinking which are not subject to opinion.
  - nwallin 7y ago
    Probably. It's possible that solving this problem in a satisfactory way would teach us something useful about number theory, that's useful in real world practical stuff. Andrew Wiles' proof of Fermat's Last Theorem is based on the modularity of elliptic curves. I'm not sure to what extent Wiles' proof contributed to elliptic curve cryptography hitting the scene a decade later, but it probably didn't hurt.
  - cyphar 7y ago
    Like a lot of these "cute" problems, finding a proof of the problem requires a new theorem or novel application of a technique which benefits mathematics overall. As a prime example, Fermat's Last Theorem was not a particularly applicable mathematical theorem. But when Andrew Wiles found a proof, he ended up proving several other elliptic curve conjectures which were important to the field.
scrungus 7y ago
33 and 42 are well known as some of the most important numbers in all of mathematics, transcending multiple fields. https://en.wikipedia.org/wiki/33_(number) https://en.wikipedia.org/wiki/33_(number) https://en.wikipedia.org/wiki/42_(number) https://en.wikipedia.org/wiki/42_(number)
- arbitrage 7y ago
  numerology is not mathematics.
  - seph-reed 7y ago
    While I do agree with you, I wouldn't be so quick to cast out numerology as a thing of importance. Hippies are pretty decent at sniffing things out, even if they explain them in totally insane ways. EDIT: If you're opposed to this comment, I request that you explain your reasoning. Making the assumption that happiness is of any value, there's actually quite a bit to dig into here.
    - cyphar 7y ago
      Numerology is factually not a branch of mathematics. As for "hippies are pretty decent at sniffing things out", just because someone was correct about something in the past (despite a lack of evidence) does not make them correct today if they still lack evidence. The burden of proof is on numerologists to show that they are correct, not on everyone else to disprove it.
      - perl4ever 7y ago
        What is the reason for bringing up numerology? It was apparently prompted by the Wikipedia links, but I don't see why.
        cyphar 7y ago
        The comment further up the chain (with the wikipedia links) was referencing the "importance" of 33 and 42 because they show up in quite a few fields -- assigning importance to certain numerical values is a kind of watered-down numerology (the fact that 42 was interesting in this problem doesn't necessarily mean that its "interesting-ness" transfers to the number itself). (But I didn't bring it up, someone else did -- I was just responding to the argument that "there might be something to it".)
eino 7y ago
It was the smallest number for which we didn't have an answer. A few months ago it was 33 (https://www.quantamagazine.org/sum-of-three-cubes-problem-solved-for-stubborn-number-33-20190326/ https://www.quantamagazine.org/sum-of-three-cubes-problem-so...), and it seems it's now 114 (https://twitter.com/robinhouston/status/1169878224416829442 https://twitter.com/robinhouston/status/1169878224416829442)
eldavojohn 7y ago
https://www.quantamagazine.org/sum-of-three-cubes-problem-solved-for-stubborn-number-33-20190326/ https://www.quantamagazine.org/sum-of-three-cubes-problem-so... Was featured on HN earlier: https://news.ycombinator.com/item?id=19492091 https://news.ycombinator.com/item?id=19492091 Lot of context there.
Someone 7y ago
It’s a problem that is easy to state, yet hard to fully solve, and doesn’t seem like it should be unsurmountable, as it is easy to make some progress on (the “integers that equal 4 or 5 modulo 9 can’t be written as the sum of 3 powers of integers” part handles 22% of cases, and interested kids can understand it) That makes it interesting to most mathematicians (if it weren’t easy to state, it could still be interesting, but to a smaller audience) Edit: it also is easy for non-mathematicians with a knack for computing to get a crack at. You need not know much about number theory to write a program that searches for solutions for the unknown cases (it will help, but knowing how to optimize programs may help more, and that’s not something all mathematicians are good in). That grows the set of people who might find this interesting.
nilkn 7y ago
There's a large gulf between the simplicity of the question and the difficulty of obtaining the answer. Whenever a simple question cannot be answered via any known simple method, it seems as if one has unexpectedly come up against something deeper and more fundamental.

Numberphile is a great resource for pretty much any "pop math" topic. https://www.youtube.com/watch?v=ASoz_NuIvP0 https://www.youtube.com/watch?v=ASoz_NuIvP0

ArneVogel 7y ago
They also just released a video on 42: https://www.youtube.com/watch?v=zyG8Vlw5aAw https://www.youtube.com/watch?v=zyG8Vlw5aAw

ajross 7y ago

The form of the statement sounds like something related to Fermat's last theorem, I guess?

fl0wenol 7y ago

It is an open question as to whether every integer not equal to 4 or 5 modulo 9 is the sum of three cubes, although it is suspected to be true. https://en.wikipedia.org/wiki/Sums_of_three_cubes#Computational_results https://en.wikipedia.org/wiki/Sums_of_three_cubes#Computatio... 33 and 42 were known to be exceptions of all sums less than 100 for which solutions were found, until recently. 42 was the most recent to fall.

huntermonk 7y ago
Thanks, this is helpful. Why do we care if every integer not equal to 4 or 5 modulo 9 is the sum of three cubes? Just for fun?
- ipince 7y ago
  Well, yeah, pretty much. In number theory you study properties/patterns of the numbers. This is an example of such a property. These properties often seem like they don’t have any applications, and often they don’t, but not always. For example, cryptography is basically all based on number theory.
  - seph-reed 7y ago
    I watched a great series once on generating ~prime numbers which explained how to use number theory to check a randomly generated number for a probability of being prime. By using a bunch of weird rules, you can be ~99% sure it's prime without having to do huge computations.
- tejtm 7y ago
  Some folks are attracted to ways of thinking which are not subject to opinion.
- nwallin 7y ago
  Probably. It's possible that solving this problem in a satisfactory way would teach us something useful about number theory, that's useful in real world practical stuff. Andrew Wiles' proof of Fermat's Last Theorem is based on the modularity of elliptic curves. I'm not sure to what extent Wiles' proof contributed to elliptic curve cryptography hitting the scene a decade later, but it probably didn't hurt.
- cyphar 7y ago
  Like a lot of these "cute" problems, finding a proof of the problem requires a new theorem or novel application of a technique which benefits mathematics overall. As a prime example, Fermat's Last Theorem was not a particularly applicable mathematical theorem. But when Andrew Wiles found a proof, he ended up proving several other elliptic curve conjectures which were important to the field.

scrungus 7y ago

33 and 42 are well known as some of the most important numbers in all of mathematics, transcending multiple fields. https://en.wikipedia.org/wiki/33_(number) https://en.wikipedia.org/wiki/33_(number) https://en.wikipedia.org/wiki/42_(number) https://en.wikipedia.org/wiki/42_(number)

arbitrage 7y ago
numerology is not mathematics.
- seph-reed 7y ago
  While I do agree with you, I wouldn't be so quick to cast out numerology as a thing of importance. Hippies are pretty decent at sniffing things out, even if they explain them in totally insane ways. EDIT: If you're opposed to this comment, I request that you explain your reasoning. Making the assumption that happiness is of any value, there's actually quite a bit to dig into here.
  - cyphar 7y ago
    Numerology is factually not a branch of mathematics. As for "hippies are pretty decent at sniffing things out", just because someone was correct about something in the past (despite a lack of evidence) does not make them correct today if they still lack evidence. The burden of proof is on numerologists to show that they are correct, not on everyone else to disprove it.
    - perl4ever 7y ago
      What is the reason for bringing up numerology? It was apparently prompted by the Wikipedia links, but I don't see why.
      - cyphar 7y ago
        The comment further up the chain (with the wikipedia links) was referencing the "importance" of 33 and 42 because they show up in quite a few fields -- assigning importance to certain numerical values is a kind of watered-down numerology (the fact that 42 was interesting in this problem doesn't necessarily mean that its "interesting-ness" transfers to the number itself). (But I didn't bring it up, someone else did -- I was just responding to the argument that "there might be something to it".)

eino 7y ago

It was the smallest number for which we didn't have an answer. A few months ago it was 33 (https://www.quantamagazine.org/sum-of-three-cubes-problem-solved-for-stubborn-number-33-20190326/ https://www.quantamagazine.org/sum-of-three-cubes-problem-so...), and it seems it's now 114 (https://twitter.com/robinhouston/status/1169878224416829442 https://twitter.com/robinhouston/status/1169878224416829442)

eldavojohn 7y ago

https://www.quantamagazine.org/sum-of-three-cubes-problem-solved-for-stubborn-number-33-20190326/ https://www.quantamagazine.org/sum-of-three-cubes-problem-so... Was featured on HN earlier: https://news.ycombinator.com/item?id=19492091 https://news.ycombinator.com/item?id=19492091 Lot of context there.

Someone 7y ago

It’s a problem that is easy to state, yet hard to fully solve, and doesn’t seem like it should be unsurmountable, as it is easy to make some progress on (the “integers that equal 4 or 5 modulo 9 can’t be written as the sum of 3 powers of integers” part handles 22% of cases, and interested kids can understand it) That makes it interesting to most mathematicians (if it weren’t easy to state, it could still be interesting, but to a smaller audience) Edit: it also is easy for non-mathematicians with a knack for computing to get a crack at. You need not know much about number theory to write a program that searches for solutions for the unknown cases (it will help, but knowing how to optimize programs may help more, and that’s not something all mathematicians are good in). That grows the set of people who might find this interesting.

nilkn 7y ago

There's a large gulf between the simplicity of the question and the difficulty of obtaining the answer. Whenever a simple question cannot be answered via any known simple method, it seems as if one has unexpectedly come up against something deeper and more fundamental.