Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

A Strange Grid Reveals Hidden Connections Between Simple Numbers

quantamagazine.org

Post reply on HN
Print view View original

11–20 of 40 posts

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#11

Post by desc »

My intuition was already screaming 'it'll be the other way around for a geometric progression' before I read that far, but I'm damned if I can understand or even speculate why. Most likely I don't properly understand it.

I'm inclined to wonder if there's a third operator which could be tested like this, such as exponentiation, but that's not commutative over integers.

Of course, if there are similarly intriguing patterns for noncommutative operators (and their sequences) the obvious next step would be to look at complex numbers and quaternions...

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#13

Post by phkahler »

zem wrote: post #12
pwaai wrote: post #8

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

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.

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.

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#14

Post by CamperBob2 »

Someone wrote: post #7
CamperBob2 wrote: post #2

Maybe I don't understand the full implications of the problem, but it seems obvious that a multiplicative table will result in more 'numerical diversity' than a summation table. Both multiplication and addition yield points on lines described by the usual mx+b equation. Addition constrains x to 1, so all of the results will be clustered near the origin, with more coincidences as an unavoidable result. Conversely, if…

”but it seems obvious that a multiplicative table will result in more 'numerical diversity' than a summation table” That isn’t true. If you pick the set of numbers {1, 2, 4, 8} , you get 10 different sums (2, 3, 4, 5, 6, 8, 9, 10, 12, and 16) but only 7 different products (1, 2, 4, 8, 16, 32, and 64)

Sure, you can always find special-case exceptions, but I still don't see why the result is surprising in the general case. There are a lot more arithmetic progressions than geometric ones.

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#15

Post by AnimalMuppet »

phkahler wrote: post #13
zem wrote: post #12

Earlier quoted context omitted.

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.

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.

Like, by picking numbers that are neither an arithmetic progression nor a geometric progression.

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#16

Post by empath75 »

CamperBob2 wrote: post #14
Someone wrote: post #7

Earlier quoted context omitted.

”but it seems obvious that a multiplicative table will result in more 'numerical diversity' than a summation table” That isn’t true. If you pick the set of numbers {1, 2, 4, 8} , you get 10 different sums (2, 3, 4, 5, 6, 8, 9, 10, 12, and 16) but only 7 different products (1, 2, 4, 8, 16, 32, and 64)

Sure, you can always find special-case exceptions, but I still don't see why the result is surprising in the general case. There are a lot more arithmetic progressions than geometric ones.

They’re both countably infinite.

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#17

Post by AnimalMuppet »

CamperBob2 wrote: post #14
Someone wrote: post #7

Earlier quoted context omitted.

”but it seems obvious that a multiplicative table will result in more 'numerical diversity' than a summation table” That isn’t true. If you pick the set of numbers {1, 2, 4, 8} , you get 10 different sums (2, 3, 4, 5, 6, 8, 9, 10, 12, and 16) but only 7 different products (1, 2, 4, 8, 16, 32, and 64)

Sure, you can always find special-case exceptions, but I still don't see why the result is surprising in the general case. There are a lot more arithmetic progressions than geometric ones.

> There are a lot more arithmetic progressions than geometric ones.

I'm pretty sure that's not true. An arithmetic progression is x[n] = x[n-1] + k and a geometric progression is x[n] = x[n-1] * k

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#18

Post by sn41 »

CamperBob2 wrote: post #14
Someone wrote: post #7

Earlier quoted context omitted.

”but it seems obvious that a multiplicative table will result in more 'numerical diversity' than a summation table” That isn’t true. If you pick the set of numbers {1, 2, 4, 8} , you get 10 different sums (2, 3, 4, 5, 6, 8, 9, 10, 12, and 16) but only 7 different products (1, 2, 4, 8, 16, 32, and 64)

Sure, you can always find special-case exceptions, but I still don't see why the result is surprising in the general case. There are a lot more arithmetic progressions than geometric ones.

In some deep sense, what you are saying is true : for example, if an infinite sequence of numbers is sufficiently dense, then it has arbitrarily long arithmetic progressions.[1] On the other hand, Rankin in 1990 constructed a set with positive density which does not have long geometric progressions.

[1] https://en.wikipedia.org/wiki/Szemer%C3%A9di%27s_theorem

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#19

Post by zwkrt »

pwaai wrote: post #8

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

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.

Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers

#20

Post by CamperBob2 »

AnimalMuppet wrote: post #17
CamperBob2 wrote: post #14

Earlier quoted context omitted.

Sure, you can always find special-case exceptions, but I still don't see why the result is surprising in the general case. There are a lot more arithmetic progressions than geometric ones.

> There are a lot more arithmetic progressions than geometric ones. I'm pretty sure that's not true. An arithmetic progression is x[n] = x[n-1] + k and a geometric progression is x[n] = x[n-1] * k

That's a good point, I'll have to stand corrected on that. Just a matter of swapping one operation for another.
Post reply on HN