3 ms·

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

by 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}.

by pavas 6y ago

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}.