more info on how I happened on them in https://math.stackexchange.com/questions/2899878 Wouter. -----Original Message----- From: James Propp Sent: Thursday, August 30, 2018 11:25 PM To: Dan Asimov ; math-fun Subject: Re: [math-fun] integers that are either square or thrice a square, where do such beasts hide? Maybe someone else has already pointed this out (or maybe everyone thinks it’s too obvious to bear pointing out), but the original question involved numbers that are either a square *or* thrice a square, not numbers that are a square *plus* thrice a square. (I’ve frequently encountered the set of numbers that are either square or pronic {1,2,4,6,9,12,16,20,...}, but that’s neither here nor there.) Jim Propp On Thursday, August 30, 2018, Dan Asimov <dasimov@earthlink.net> wrote:

Thanks for the explanations — how cool!

—Dan

----- From memory, as the drawing is still in some box (I moved):

I find squares on the x-axis and the 6 equivalent axes

(repeatedly multiply by w_6).

I find 3 * squares on the axis along w_{12} and the 6 equivalent axes (repeatedly multiply by w_6).

Best regards, jj

* Adam P. Goucher <apgoucher@gmx.com> [Aug 30. 2018 17:14]:

...

I'm using it to mean "arg(z) is a rational multiple of 2 pi", or equivalently "some power of z is real" or "z / abs(z) is a root of unity".

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