set $\left\{\left|a + b\omega + c\omega^{2}\right|^{2} : a,b,c \text{ distinct non-zero integers}\right\}$ equals $\underline{\hspace{2cm}}$.

set {|a + bω + cω²|² : a,b,c distinct non-zero integers} equals underlinehspace2cm.

$\omega^3 = 1, \quad 1 + \omega + \omega^2 = 0$

$a,\ b,\ c \text{ distinct non-zero integers}$

$z = a + b\omega + c\omega^{2}$

$\text{To find: the set of all possible values of } |z|^{2}$

$1.\ |z|^{2} \text{ via } z\bar z$

$2.\ \text{use } 1+\omega+\omega^2=0$

$3.\ \text{sum of squares; write the set}$

$1 + \omega + \omega^2 = 0$

$|z|^{2} = z\bar{z}, \quad \bar{z} = a + b\omega^{2} + c\omega$

$|z|^{2} = (a+b\omega+c\omega^{2})(a+b\omega^{2}+c\omega)$

$a\cdot a = a^2,\quad (b\omega)(b\omega^2) = b^2\omega^3 = b^2,\quad (c\omega^2)(c\omega) = c^2\omega^3 = c^2$

$\text{cross terms} = (ab+bc+ca)(\omega+\omega^2)$

$|z|^{2} = a^2+b^2+c^2 - ab - bc - ca$

$|z|^{2} = \tfrac{1}{2}\left[(a-b)^2+(b-c)^2+(c-a)^2\right]$

$a,b,c \text{ distinct} \Rightarrow (a-b),(b-c),(c-a) \neq 0$ $(a-b)+(b-c)+(c-a)=0$

$\min = \tfrac{1}{2}(1+1+4) = 3 \quad (\text{e.g. } a,b,c = 2,3,4)$

$(a,b,c)=(1,2,4):\ |z|^{2} = \tfrac{1}{2}(1+4+9) = 7$

$|z|^{2} \equiv a^2+b^2+c^2+2(ab+bc+ca) = (a+b+c)^2 \pmod 3$

$\Rightarrow |z|^{2} \equiv 0 \text{ or } 1 \pmod 3$

$x=a-b,\ y=b-c \Rightarrow |z|^{2} = x^2+xy+y^2$

$xy(x+y) \neq 0$

$\text{any } x^2+xy+y^2,\ xy(x+y)\neq0, \text{ is attainable (choose } c \text{ large)}$

$\{3,\,7,\,12,\,13,\,19,\,21,\,27,\,28,\,\dots\}$

$\boxed{\left\{\,x^2+xy+y^2 : x,y \in \mathbb{Z},\ xy(x+y) \neq 0\,\right\}}$