Since Q(zeta), where zeta is a primitive 3rd root of unity has class number 1, the situation as to whether an integer is of the form x^2 + xy + y^2 is similar to the situation with x^2 + y^2: n is of the that form if and only if every prime p dividing n which is = 5 mod 6 divides it to an even power. The density of of 1/sqrt(x) that Rich mentioned is an old result due to Landau. Victor On Wed, Jul 20, 2016 at 4:11 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Numbers of the form K^2 + KL +L^2, K and L integers, arise often when making calculations connected with the triangular lattice in the plane.
Let tau = exp(2pi*i/6).
For K, L > 0, K^2 + KL +L^2 is, among other things, the number of points in the quotient ring
Z{tau] / <K + L*tau>
namely, the Eisenstein integers factored out by its ideal <K + L*tau>.
I would like to know more about these numbers.
For example:
* Given any integer N, is there a simple test for whether N is of the form K^2 + KL + L^2 for some integers K, L ???
* What can be said about the prime factorization of K^2 + KL + L^2 ???
* Is there a relationship of such numbers to Eisenstein primes?
—Dan
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