5 ms·

summarised as: yeah no. What about complex numbers? is there something similar there? Not even so much necessarily on the negative numbers. Anything interestin

summarised as: yeah no. What about complex numbers? is there something similar there? Not even so much necessarily on the negative numbers. Anything interesting amongst the Js?

pmiller2 6y ago
Yes, there is such a thing as complex prime numbers. If you restrict yourself to looking at numbers whose real and imaginary parts are both integers, then you have a ring called the Gaussian integers. A Gaussian integer a + bi is prime if and only if neither a nor b are zero and a^2 + b^2 is a prime number in the integers, or it is a product of +-1 or +-i and a prime integer of the form 4k+3. Rabbit hole begins here: https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_primes https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_prim...
- sfgweilr4f 6y ago
  There goes my weekend in a mathematical whoosh.
  - pmiller2 6y ago
    You're welcome. ;)
- lrossi 6y ago
  Prime numbers in 2 dimensions. I’ve never thought of this before, but I like the idea. Interestingly, when plotting them, some patterns appear: https://demonstrations.wolfram.com/GaussianPrimes/ https://demonstrations.wolfram.com/GaussianPrimes/ I would have expected patterns due to symmetry around the axes, but there seems to be more than that.
  - pmiller2 6y ago
    If you like that, then you'll love this: https://en.wikipedia.org/wiki/Ulam_spiral https://en.wikipedia.org/wiki/Ulam_spiral
    - edoceo 6y ago
      I did! Thanks¡
    - akhann 6y ago
      Oh cool, thanks for posting this! I used to do that back in school to kill time and later started using it to create drawings by following connect-the-dots rules. We had a get together some time ago (before Covid) where we did communal painting, and one of the rules produced a nice bird-looking thing. :) Great to see there is a name for this sort of thing.
  - ogogmad 6y ago
    And 4 dimensions: See Hurwitz quaternions.
    - pmiller2 6y ago
      Huh, I'd never heard of that one. I suppose that means that E8 is the equivalent for octonions? That loses the concept of "prime," which is no surprise, because the octonions are not a field, but I wonder if one could bring it back by substituting irreducible representations for primes, somehow? I'm not sure how to carry this off, because I dropped representation theory in grad school, but it seems like an interesting concept to play with, at least.
  - Sharlin 6y ago
    There are patterns along the coordinate axes and also along the diagonals, but otherwise the distribution looks pretty random: https://commons.wikimedia.org/wiki/File:Gauss-primes-768x768.png https://commons.wikimedia.org/wiki/File:Gauss-primes-768x768...

summarised as: yeah no. What about complex numbers? is there something similar there? Not even so much necessarily on the negative numbers. Anything interesting amongst the Js?

pmiller2 6y ago

Yes, there is such a thing as complex prime numbers. If you restrict yourself to looking at numbers whose real and imaginary parts are both integers, then you have a ring called the Gaussian integers. A Gaussian integer a + bi is prime if and only if neither a nor b are zero and a^2 + b^2 is a prime number in the integers, or it is a product of +-1 or +-i and a prime integer of the form 4k+3. Rabbit hole begins here: https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_primes https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_prim...

sfgweilr4f 6y ago
There goes my weekend in a mathematical whoosh.
- pmiller2 6y ago
  You're welcome. ;)
lrossi 6y ago
Prime numbers in 2 dimensions. I’ve never thought of this before, but I like the idea. Interestingly, when plotting them, some patterns appear: https://demonstrations.wolfram.com/GaussianPrimes/ https://demonstrations.wolfram.com/GaussianPrimes/ I would have expected patterns due to symmetry around the axes, but there seems to be more than that.
- pmiller2 6y ago
  If you like that, then you'll love this: https://en.wikipedia.org/wiki/Ulam_spiral https://en.wikipedia.org/wiki/Ulam_spiral
  - edoceo 6y ago
    I did! Thanks¡
  - akhann 6y ago
    Oh cool, thanks for posting this! I used to do that back in school to kill time and later started using it to create drawings by following connect-the-dots rules. We had a get together some time ago (before Covid) where we did communal painting, and one of the rules produced a nice bird-looking thing. :) Great to see there is a name for this sort of thing.
- ogogmad 6y ago
  And 4 dimensions: See Hurwitz quaternions.
  - pmiller2 6y ago
    Huh, I'd never heard of that one. I suppose that means that E8 is the equivalent for octonions? That loses the concept of "prime," which is no surprise, because the octonions are not a field, but I wonder if one could bring it back by substituting irreducible representations for primes, somehow? I'm not sure how to carry this off, because I dropped representation theory in grad school, but it seems like an interesting concept to play with, at least.
- Sharlin 6y ago
  There are patterns along the coordinate axes and also along the diagonals, but otherwise the distribution looks pretty random: https://commons.wikimedia.org/wiki/File:Gauss-primes-768x768.png https://commons.wikimedia.org/wiki/File:Gauss-primes-768x768...