Dan Asimov <dasimov@earthlink.net> wrote:
P.S. Now I'd like to know the same thing for the Gaussian integers Z[i] and the Eisenstein integers Z[w] where w = exp(2pi*i/3). These apparently have some complicated factor rings.
I'm not sure what you mean by modular arithmetic on Gaussian integers. * Modular in both real and imaginary, meaning that the complex plane is tessellated with rectangles of some integer side lengths, and the rectangles are all identified with each other? Equivalently, there's just one rectangle, and its opposite edges are connected, making a torus? * Modular just in the reals, with the imaginaries unbounded? * Modular just in the imaginaries, with the reals unbounded? I think modular in both real and imaginary sounds most interesting. Especially since if you think of it as a tessellation of rectangles, you can generalize to offset rectangles, like bricks in a wall. And if you think of it as a torus, you can generalize to a Klein bottle. (I wonder what the Mandelbrot set looks like in the modular complex plane.) Long ago I noticed that modulo a number that's one more than a square, negative one has a real square root. For instance mod 10, i = 3 and -i = 7. (Or vice versa if you prefer.) All Gaussian integers have consistent real integer values mod 10. For instance 2i is 6, 2i-5 is 1, etc. But this doesn't seem to get you anything new and interesting. I played around a lot with Gaussian integers. I calculated and plotted, by hand, the first hundred or so Gaussian primes. I also calculated and plotted Gaussian squares, and noticed that they formed interlocking parabolas. I never played with Eisenstein integers. Would you use a triangular tessellation for them when going modular? Are those the only two interesting ways of generalizing integers to the complex plane? What about the vertices of a regular hexagonal tessellation?