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/
I would have expected patterns due to symmetry around the axes, but there seems to be more than that.
If you like that, then you'll love this: https://en.wikipedia.org/wiki/Ulam_spiral
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.
I did! Thanks¡
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...
And 4 dimensions: See Hurwitz quaternions.
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.
Re:Do we have negative prime numbers?
by sfgweilr4f on Saturday January 16, 2021 @10:51AM
(#25802977)
There goes my weekend in a mathematical whoosh.
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_prim...