6 ms·

The Prime Hexagon

36 points by aww_dang 6y ago

dcow 6y ago
This is from 2016. Has anything further insight been gleaned from this interesting property?
- bombcar 6y ago
  It's been updated since then at least - there's a note that the sidebar doesn't work sometimes (2018).
  - pstoll 6y ago
    He’s still working on it.
- klyrs 6y ago
  Mathematically, it's not terribly deep -- at risk of being glib, it essentially follows from 2 and 3 being the only primes divisible by 2 or 3. It's pretty cool, but I wouldn't expect deep insights coming from this.
pavas 6y ago
Am I missing something here? Choose any infinite sequence of binary numbers {0,1,0,0,...} and map the natural numbers to it. For any number not divisible by 2 or 3, go left if it maps to 0 and right if it maps to 1. Doesn't this property still hold? That is, it's nothing to do with prime numbers and something to do with the quotient of {2,3}.
- fogof 6y ago
  Yeah I think the hexagon property holds whenever you have a set which consists of 1 and 5 mod 6 numbers.
- AnotherGoodName 6y ago
  Yep. All prime numbers above 6 are of the form 6n + 1 or 6n + 5, everything else is a factor of 2 or 3. If you make a choice that occurs on primes then no choice will be made for the 6n + 2/3/4 cells. This gives a well defined pattern since this is setup so those no choice cells are the ones that could go outside the bounds. You could also create a shape with 30 sides and a similar pattern since all primes above 30 (235) are of the form 30n + 1, 30n + 7, 30n + 11, 30n + 13, 30n + 17, 30n + 19, 30n + 23 or 30n + 29. Everything else is divisible by 2, 3 or 5. In fact you can do this sort of thing with any set of factors. There will be regular gaps in primality. Set the starting point so that those gaps in primality only change the direction on the inward sides and you'll confine all numbers within some larger shape.
  - mkl 6y ago
    > 235 You can avoid HN's * = italics by using spaces (2 * 3 * 5) or escaping the * with \ (2\*3\*5 gives 2*3*5).
  - lupire 6y ago
    3B1B has a semi related video on how working with prime numbers can help you see properties that are also found in composite numbers, since the sequence of primes overlaps with many other sequences. https://m.youtube.com/watch?v=EK32jo7i5LQ https://m.youtube.com/watch?v=EK32jo7i5LQ
  - pavas 6y ago
    You've given a much better explanation of it than I have. I was trying to think if there is some interesting property of the generalization of this (generalize the quotients/generalize the directions) or if it is all just a trivial consequence of primality. I guess the generalization here is choose a set {primes}, create a shape (I believe that would be called a prime lattice or something?) w/ Product({primes}) sides and all primes > Product({primes}) are of the form {Product({primes}) + Set_of_Primes/{primes} mod Product({primes})}. I'm not sure how to get the inductive proof that these moves converge within the lattice for all {primes}.
gnulinux 6y ago
I don't understand why this is in HN frontpage. This is just numerology. There is no predictive or mathematical content here, right? Am I wrong?
- pstoll 6y ago
  Disclaimer - friend of the author and have done the programming for some of his research. (Now I wish I’d cleaned it up more but oh well...) I’d describe him as a hobby mathematician, not formally trained. But I wouldn’t put it in the numerology realm.
Flocular 6y ago
The only surprising thing would be the pi-numbers. However the statistical evidence there isnt very strong. It's quite likely he looked at more than 400 series, so finding a 1 in 400-pattern in one of them is no surprise.
- andrewflnr 6y ago
  I'm inclined to think it's boring in the opposite direction. Pi is everywhere. It would be more surprising if there was no link to pi.
smoldesu 6y ago
I'm pleasantly surprised, any time I see a *spin.com website I immediately fear for my life and grimace as I click on the link.
deleted 6y ago
[deleted]
andrewflnr 6y ago
For IMO a more meaningful phenomenon, this video by 3Blue1Brown explains why primes plotted as (p, p) in polar coordinates produce interesting patterns. https://youtu.be/EK32jo7i5LQ https://youtu.be/EK32jo7i5LQ

The Prime Hexagon

36 points by aww_dang 6y ago

dcow 6y ago

This is from 2016. Has anything further insight been gleaned from this interesting property?

bombcar 6y ago
It's been updated since then at least - there's a note that the sidebar doesn't work sometimes (2018).
- pstoll 6y ago
  He’s still working on it.
klyrs 6y ago
Mathematically, it's not terribly deep -- at risk of being glib, it essentially follows from 2 and 3 being the only primes divisible by 2 or 3. It's pretty cool, but I wouldn't expect deep insights coming from this.

pavas 6y ago

Am I missing something here? Choose any infinite sequence of binary numbers {0,1,0,0,...} and map the natural numbers to it. For any number not divisible by 2 or 3, go left if it maps to 0 and right if it maps to 1. Doesn't this property still hold? That is, it's nothing to do with prime numbers and something to do with the quotient of {2,3}.

fogof 6y ago
Yeah I think the hexagon property holds whenever you have a set which consists of 1 and 5 mod 6 numbers.
AnotherGoodName 6y ago
Yep. All prime numbers above 6 are of the form 6n + 1 or 6n + 5, everything else is a factor of 2 or 3. If you make a choice that occurs on primes then no choice will be made for the 6n + 2/3/4 cells. This gives a well defined pattern since this is setup so those no choice cells are the ones that could go outside the bounds. You could also create a shape with 30 sides and a similar pattern since all primes above 30 (235) are of the form 30n + 1, 30n + 7, 30n + 11, 30n + 13, 30n + 17, 30n + 19, 30n + 23 or 30n + 29. Everything else is divisible by 2, 3 or 5. In fact you can do this sort of thing with any set of factors. There will be regular gaps in primality. Set the starting point so that those gaps in primality only change the direction on the inward sides and you'll confine all numbers within some larger shape.
- mkl 6y ago
  > 235 You can avoid HN's * = italics by using spaces (2 * 3 * 5) or escaping the * with \ (2\*3\*5 gives 2*3*5).
- lupire 6y ago
  3B1B has a semi related video on how working with prime numbers can help you see properties that are also found in composite numbers, since the sequence of primes overlaps with many other sequences. https://m.youtube.com/watch?v=EK32jo7i5LQ https://m.youtube.com/watch?v=EK32jo7i5LQ
- pavas 6y ago
  You've given a much better explanation of it than I have. I was trying to think if there is some interesting property of the generalization of this (generalize the quotients/generalize the directions) or if it is all just a trivial consequence of primality. I guess the generalization here is choose a set {primes}, create a shape (I believe that would be called a prime lattice or something?) w/ Product({primes}) sides and all primes > Product({primes}) are of the form {Product({primes}) + Set_of_Primes/{primes} mod Product({primes})}. I'm not sure how to get the inductive proof that these moves converge within the lattice for all {primes}.

gnulinux 6y ago

I don't understand why this is in HN frontpage. This is just numerology. There is no predictive or mathematical content here, right? Am I wrong?

pstoll 6y ago
Disclaimer - friend of the author and have done the programming for some of his research. (Now I wish I’d cleaned it up more but oh well...) I’d describe him as a hobby mathematician, not formally trained. But I wouldn’t put it in the numerology realm.

Flocular 6y ago

The only surprising thing would be the pi-numbers. However the statistical evidence there isnt very strong. It's quite likely he looked at more than 400 series, so finding a 1 in 400-pattern in one of them is no surprise.

andrewflnr 6y ago
I'm inclined to think it's boring in the opposite direction. Pi is everywhere. It would be more surprising if there was no link to pi.

smoldesu 6y ago

I'm pleasantly surprised, any time I see a *spin.com website I immediately fear for my life and grimace as I click on the link.

deleted 6y ago

[deleted]

andrewflnr 6y ago

For IMO a more meaningful phenomenon, this video by 3Blue1Brown explains why primes plotted as (p, p) in polar coordinates produce interesting patterns. https://youtu.be/EK32jo7i5LQ https://youtu.be/EK32jo7i5LQ