James Propp <jamespropp@gmail.com> wrote:
Is there a standard term for this tripartition?
I don't know. I discovered it independently. I think I was trying to find a parity-related proof that there was no point that has rational distances from all four corners of the unit square. I didn't succeed, and as far as I know the problem is still unsolved. No point within the plane, I mean. Outside the plane, it's easy to find such a point. But that would be a pyramid scheme. :-)
But I think there's another coloring that satisfies the conditions I stated: -r is blue if r is green and vice versa.
Maybe so. But I'm still not convinced that there isn't some completely unrelated coloring that would also solve your problem.
Keith F. Lynch <kfl@keithlynch.net> wrote:
Extra credit: Is there any color or pairs of colors that form a group under multiplication? Is there any color or any pair of colors that forms a group under addition?
Yes and yes.
Do tell.
Maybe all but finitely many convergents to pi are the same color?
Sure, and maybe all but finitely many decimal digits of pi are fours and sevens. It's never been disproven, but it seems kind of unlikely.
Maybe some quadratic irrational has the property in question, or maybe not; that might be my fall-asleep problem for tonight.
The convergents to sqrt(2) don't, but I noticed that if you cheat by starting with 2/1 instead of 1/1, and then, as always, repeatedly replacing p/q with (p+2q)/(p+q), all these pseudo-convergents will be red (even/odd), and will still converge to sqrt(2). But I'd be surprised if it's not possible to find pseudo-convergents of any color you please to converge to any real number you please. For instance one simple formula for pi is 4/1 - 4/3 + 4/5 - 4/7 + ... The partial sums of that series are all red. And the standard formula for e, the sum of the reciprocals of the factorials, gives nothing but green (odd/even) partial sums after the four four terms. Your fall-asleep problem for tonight is to find a sequence for pi whose partial sums are either all blue or all green, or a sequence for e whose partial sums are all red or all blue. :-) Is there some natural coloring for algebraic numbers? I no longer think convergents are a way to get anything interesting. Maybe something to do with the parity of the coefficients, e.g. what patterns of even and odd A, B, and C have when Ax^2 + Bx + C = 0. What are the rules for adding, subtracting, multiplying, and dividing numbers expressed in that form? I never learned it. I do know you'd typically get higher-order polynomials, e.g. the set of all quadratic numbers isn't closed under any interesting operations, similarly with the set of all cubic numbers, etc.