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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
by 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 agoThere goes my weekend in a mathematical whoosh.
- pmiller2 6y agoYou're welcome. ;)
- lrossi 6y agoPrime 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 agoIf you like that, then you'll love this: https://en.wikipedia.org/wiki/Ulam_spiral https://en.wikipedia.org/wiki/Ulam_spiral
- edoceo 6y agoI did! Thanks¡
- akhann 6y agoOh 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 agoAnd 4 dimensions: See Hurwitz quaternions.
- pmiller2 6y agoHuh, 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 agoThere 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...