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i dont get it...what is the hidden connection exactly?

by pwaai 8y ago

deleted 8y ago
[deleted]
zem 8y ago
the connection is that picking your numbers so that you end up with a lot of duplicate sums constrains you to have very few duplicate products, and vice versa.
- phkahler 8y ago
  I think it's more like having a low number of one precludes having a low number of the other. I may be wrong, but I think you can have a large nu.ber of borh.
  - AnimalMuppet 8y ago
    Like, by picking numbers that are neither an arithmetic progression nor a geometric progression.
zwkrt 8y ago
The real point is that no one gets the hidden connection. There are lots of tantalizing clues in number theory that lead us to believe there is some fundamental connection between addition and subtraction but we can't state it outright. This toy problem is like a black box in that we can play with it and quantify it's behavior but we don't know (exactly) why it works.
- qubex 8y ago
  I think you mean “addition and multiplication”, not “multiplication and subtraction”... there definitely is a lot of evidence that addition and subtraction are related, but that’s just plain trivial.
  - zwkrt 8y ago
    Thank you, mobile commenting is a blessing and a curse.
    - qubex 8y ago
      Heh, no worries. I got it wrong too... I meant to write “addition and subtraction” and ended up writing “subtraction and multiplication”... so that’s what I get for trying to be a pedant (and boy do I deserve it).
Dylan16807 8y ago
'connection' is a somewhat misleading way of stating it. It's more like the opposite of a connection. A set of numbers cannot have trait A and trait B at the same time. Trait A is that the numbers have a certain kind of similarity in their spacing, leading to few unique sums. Trait B is that they have a certain kind of similarity in their factors, leading to few unique products. So from some angle it's interesting that you can't impose both types of patterns on a single set of numbers, because that inability implies a relation between two very different operations. But on the other hand why would you assume that you should find an intersection between two restrictive algorithms? Maybe there's a hidden assumption that the algorithms would be random-ish and let you find an overlap if you search hard enough, and that assumption is screwing with people's intuition.
- function_seven 8y ago
  > Trait A is that the numbers have a certain kind of similarity in their spacing, leading to few unique sums. Trait B is that they have a certain kind of similarity in their factors, leading to few unique products. This is an excellent way to describe it. I had the sense that “of course the geometric sequences will collide more when multiplying” but if you asked me “why?” I would have trouble explaining. So thanks. As far as finding a set that has an equal number of distinct sums and products, would it be possible to come up with a sequence that’s “half”-arithmetic and “half”-geometric?
  - Dylan16807 8y ago
    Even a single value that doesn't fit in one sequence will cause a lot of extra values, so your ability to fit multiple patterns gets worse and worse as you go longer. Something like 2 4 6 8 16 does pretty well for its size but even then it's giving you twelve distinct outputs for either operation.
- feanaro 8y ago
  > But on the other hand why would you assume that you should find an intersection between two restrictive algorithms? I think it's key here that these two algorithms are so basic and fundamental to the structure they are executed upon (the naturals) so that every other algorithm will use them in some way. The very notion of "algorithm over the naturals" and intuition of what it means for such an algorithm to be restrictive are determined by the nature of these operations. Put like this, it seems far less obvious to me that there is an a priori reason why the results of these two operations should or shouldn't coincide. Both scenarios seem plausible before we study problems like these.

i dont get it...what is the hidden connection exactly?

by pwaai 8y ago

i dont get it...what is the hidden connection exactly?

deleted 8y ago

[deleted]

zem 8y ago

the connection is that picking your numbers so that you end up with a lot of duplicate sums constrains you to have very few duplicate products, and vice versa.

phkahler 8y ago
I think it's more like having a low number of one precludes having a low number of the other. I may be wrong, but I think you can have a large nu.ber of borh.
- AnimalMuppet 8y ago
  Like, by picking numbers that are neither an arithmetic progression nor a geometric progression.

zwkrt 8y ago

The real point is that no one gets the hidden connection. There are lots of tantalizing clues in number theory that lead us to believe there is some fundamental connection between addition and subtraction but we can't state it outright. This toy problem is like a black box in that we can play with it and quantify it's behavior but we don't know (exactly) why it works.

qubex 8y ago
I think you mean “addition and multiplication”, not “multiplication and subtraction”... there definitely is a lot of evidence that addition and subtraction are related, but that’s just plain trivial.
- zwkrt 8y ago
  Thank you, mobile commenting is a blessing and a curse.
  - qubex 8y ago
    Heh, no worries. I got it wrong too... I meant to write “addition and subtraction” and ended up writing “subtraction and multiplication”... so that’s what I get for trying to be a pedant (and boy do I deserve it).

Dylan16807 8y ago

'connection' is a somewhat misleading way of stating it. It's more like the opposite of a connection. A set of numbers cannot have trait A and trait B at the same time. Trait A is that the numbers have a certain kind of similarity in their spacing, leading to few unique sums. Trait B is that they have a certain kind of similarity in their factors, leading to few unique products. So from some angle it's interesting that you can't impose both types of patterns on a single set of numbers, because that inability implies a relation between two very different operations. But on the other hand why would you assume that you should find an intersection between two restrictive algorithms? Maybe there's a hidden assumption that the algorithms would be random-ish and let you find an overlap if you search hard enough, and that assumption is screwing with people's intuition.

function_seven 8y ago
> Trait A is that the numbers have a certain kind of similarity in their spacing, leading to few unique sums. Trait B is that they have a certain kind of similarity in their factors, leading to few unique products. This is an excellent way to describe it. I had the sense that “of course the geometric sequences will collide more when multiplying” but if you asked me “why?” I would have trouble explaining. So thanks. As far as finding a set that has an equal number of distinct sums and products, would it be possible to come up with a sequence that’s “half”-arithmetic and “half”-geometric?
- Dylan16807 8y ago
  Even a single value that doesn't fit in one sequence will cause a lot of extra values, so your ability to fit multiple patterns gets worse and worse as you go longer. Something like 2 4 6 8 16 does pretty well for its size but even then it's giving you twelve distinct outputs for either operation.
feanaro 8y ago
> But on the other hand why would you assume that you should find an intersection between two restrictive algorithms? I think it's key here that these two algorithms are so basic and fundamental to the structure they are executed upon (the naturals) so that every other algorithm will use them in some way. The very notion of "algorithm over the naturals" and intuition of what it means for such an algorithm to be restrictive are determined by the nature of these operations. Put like this, it seems far less obvious to me that there is an a priori reason why the results of these two operations should or shouldn't coincide. Both scenarios seem plausible before we study problems like these.