9 ms·

There's a lot of interesting structure. Does this suggest some kind of structure to the space of prime factors or is it just trying to attach meaning where none

There's a lot of interesting structure. Does this suggest some kind of structure to the space of prime factors or is it just trying to attach meaning where none exists?

adamtulinius 8y ago
"A very pretty structure emerges; this might be spurious in that it captures more about the layout algorithm than any “true” structure of numbers."
vesinisa 8y ago
Given that there is definitive visual structure to primes themselves[1], it would be rather surprising if prime factorization had none. [1] https://en.wikipedia.org/wiki/Ulam_spiral https://en.wikipedia.org/wiki/Ulam_spiral
- deleted 8y ago
  [deleted]
- tess0r 8y ago
  Ulam sprials are super interesting. I developed a visualization (and some explanation) once just to understand the problem a little better. Maybe this sparks some interest in someone :) https://tessi.github.io/walking-the-ulam-spiral/ https://tessi.github.io/walking-the-ulam-spiral/
- jwilk 8y ago
  On HN 3 months ago: https://news.ycombinator.com/item?id=17019533 https://news.ycombinator.com/item?id=17019533
thaumasiotes 8y ago
Prime factors are defined by a pretty rigid structure; you see a factor of 2 every two numbers; a factor of 3 every three numbers; a factor of 5 every five numbers...
- algorias 8y ago
  For extra fun, if you plot the number of times a specific prime factor occurs, you get a fractal pattern. E.g. for the prime factor 2: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 0 2 0 1 0 3 0 1 0 2 0 1 0
thraway180306 8y ago
structure to the space of prime factors There is topological structure https://math.stackexchange.com/questions/2879/mapping-natural-numbers-into-prime-exponents-space https://math.stackexchange.com/questions/2879/mapping-natura... http://ncatlab.org/nlab/show/arithmetic+topology http://ncatlab.org/nlab/show/arithmetic+topology as well as geometric (arithmetic geometry). Question is how does the picture correspond to any of it. Glancing at the UMAP paper https://arxiv.org/abs/1802.03426 https://arxiv.org/abs/1802.03426 it looks like a proper invariant preserving embedding, so maybe loops in the image are real. Even if not, it does hint artistically at sorts of stuff that exist.

There's a lot of interesting structure. Does this suggest some kind of structure to the space of prime factors or is it just trying to attach meaning where none exists?

adamtulinius 8y ago

"A very pretty structure emerges; this might be spurious in that it captures more about the layout algorithm than any “true” structure of numbers."

vesinisa 8y ago

Given that there is definitive visual structure to primes themselves[1], it would be rather surprising if prime factorization had none. [1] https://en.wikipedia.org/wiki/Ulam_spiral https://en.wikipedia.org/wiki/Ulam_spiral

deleted 8y ago
[deleted]
tess0r 8y ago
Ulam sprials are super interesting. I developed a visualization (and some explanation) once just to understand the problem a little better. Maybe this sparks some interest in someone :) https://tessi.github.io/walking-the-ulam-spiral/ https://tessi.github.io/walking-the-ulam-spiral/
jwilk 8y ago
On HN 3 months ago: https://news.ycombinator.com/item?id=17019533 https://news.ycombinator.com/item?id=17019533

thaumasiotes 8y ago

Prime factors are defined by a pretty rigid structure; you see a factor of 2 every two numbers; a factor of 3 every three numbers; a factor of 5 every five numbers...

algorias 8y ago
For extra fun, if you plot the number of times a specific prime factor occurs, you get a fractal pattern. E.g. for the prime factor 2: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 0 2 0 1 0 3 0 1 0 2 0 1 0

thraway180306 8y ago

structure to the space of prime factors There is topological structure https://math.stackexchange.com/questions/2879/mapping-natural-numbers-into-prime-exponents-space https://math.stackexchange.com/questions/2879/mapping-natura... http://ncatlab.org/nlab/show/arithmetic+topology http://ncatlab.org/nlab/show/arithmetic+topology as well as geometric (arithmetic geometry). Question is how does the picture correspond to any of it. Glancing at the UMAP paper https://arxiv.org/abs/1802.03426 https://arxiv.org/abs/1802.03426 it looks like a proper invariant preserving embedding, so maybe loops in the image are real. Even if not, it does hint artistically at sorts of stuff that exist.