Hi, Keith. On Sunday, April 17, 2016, Keith F. Lynch <kfl@keithlynch.net <javascript:_e(%7B%7D,'cvml','kfl@keithlynch.net');>> wrote:
Argh: 355/113 isn't in [0,1]; replace it by 113/355.
Either way, it's blue.
Odd/odd is blue. Even/odd is red. Odd/even is green. Even/even can be reduced to one of the above. I have no idea how to prove that this coloring is unique, however. Do you have a proof?
There are two issues: existence and uniqueness. For existence, one colors the rationals as Keith says. One must then show that if a/b and c/d are fractions with ad-bc=1, the fraction a/b, the fraction c/d, and their mediant (a+c)/(b+d) must have different colors; simple consideration of cases suffices. For uniqueness, one may cite the theory of Farey fractions and the Stern-Brocot tree. Every rational can be obtained by an iterative process of inserting mediants, starting from 0/1 and 1/1. This means that at each stage one has no choice (that is to say, that one has 1 choice).
I noticed, years ago, that the rationals have these three parities, so that was the first thing I tried.
Is there a standard term for this tripartition? Note that it can be described as the sign of the 2-adic valuation.
I agree with RCS that this is unambiguous over the whole of the rationals. I don't understand why you think there's any ambiguity with negatives. Just as negative integers have the same parity as if they were positive, so do negative rationals.
Your notion of "parity" is unambiguous, and indeed natural, but my question was about 3-colorings. I may be mistaken, but I think the uniqueness proof fails if one tri-colors all the rationals. If you disagree, please show me how to deduce the color of -1. Just to be clear: I agree completely that the natural 3-coloring assigns -r the same color as r, for all r. But I think there's another coloring that satisfies the conditions I stated: -r is blue if r is green and vice versa.
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.
Can this coloring be extended in any natural way to any irrationals? You mention 355/113, which is best known as one of the convergents to pi. Are there any irrational numbers whose convergents are all the same color? (Pi isn't one.)
Maybe all but finitely many convergents to pi are the same color? If so, it might make sense to regard pi as having that color. My guess is that (a) pi shows no signs of having this property, and (b) we have no idea for how to approach the question. Maybe some quadratic irrational has the property in question, or maybe not; that might be my fall-asleep problem for tonight. (Is that what Charles Dodgson meant by a "pillow problem"?) Jim Propp