Re: [math-fun] integers that are either square or thrice a square, where do such beasts hide?
With "rational arguments" — not sure I know what that means. Eisenstein integers should be the ring Z[w] where w = -1/2 + i*sqrt(3/4). Since Z[w] = {K + L*w | K, L in Z} we can compute |K + L*w|^2 = (K - L/2)^2 + (3/4)*L^2 = K^2 - KL + L^2 (which is what I remembered). I'm still in the dark over what rational arguments are, and how K^2 - KL + L^2 squares with what Adam wrote. —Dan Adam Goucher wrote: ----- They're precisely the norms of Eisenstein integers with rational arguments. Wouter Meeussen wrote: -----
do they occur anywhere naturally? in combinatorics or in number theory? I bumped into them by accident looking at bracelet-stuff.
-----
* Dan Asimov <dasimov@earthlink.net> [Aug 30. 2018 08:41]:
With "rational arguments" — not sure I know what that means.
The set of integers of the forms x^2 - x * y + y^2 (using [0,1] and w_3), x^2 + x * y + y^2 (using [0,1] and w_6), and x^2 + 3 * y^2 (using ???) are all the same (norms of Eisensteins). In the last form, with either x or y zero, one gets what was asked for. Not sure that is what 'rational' means here (with Gaussian, 'rational' being the 4k+3 primes, those with zero imaginary part). Best regards, jj
[...]
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".
Sent: Wednesday, August 29, 2018 at 8:47 PM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] integers that are either square or thrice a square, where do such beasts hide?
With "rational arguments" — not sure I know what that means.
Eisenstein integers should be the ring Z[w] where w = -1/2 + i*sqrt(3/4).
Since Z[w] = {K + L*w | K, L in Z} we can compute
|K + L*w|^2 = (K - L/2)^2 + (3/4)*L^2
= K^2 - KL + L^2
(which is what I remembered).
I'm still in the dark over what rational arguments are, and how K^2 - KL + L^2 squares with what Adam wrote.
—Dan
Adam Goucher wrote: ----- They're precisely the norms of Eisenstein integers with rational arguments.
Wouter Meeussen wrote: -----
do they occur anywhere naturally? in combinatorics or in number theory? I bumped into them by accident looking at bracelet-stuff.
-----
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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".
[...]
participants (3)
-
Adam P. Goucher -
Dan Asimov -
Joerg Arndt