О делении с остатком в одном неевклидовом кольце

В книге "Алгебра" ван дер Вардена есть следующая задача.

В кольце чисел $a+b\sqrt{-3}$, где $a$ и $b$ --- целые числа (мы будем обозначать это кольцо через $R$), число 4 разлагается на простые множители двумя существенно различными способами:
$$
4 = 2\cdot 2=(1+\sqrt{-3})(1-\sqrt{-3}).
$$

Это значит, что кольцо $R$ не является евклидовым. Невозможно определить для $R$ норму и деление с остатком, удовлетворящие определению евклидова кольца.

Определим норму числа $\alpha= a+b\sqrt{-3}$:
$$
N(\alpha) = a^2+3b^2.
$$
Это просто квадрат модуля соответствующего комплексного числа. Поэтому
$$
N(\alpha\beta)=N(\alpha)N(\beta).
$$

Пусть $\alpha,\beta\in R, \ \beta\neq 0$. Так же, как и для целых гауссовых чисел, найдём такое дробное число $\lambda'$, принадлежащее полю частных кольца $R$, что $\alpha = \beta\lambda'$.
$$
\lambda'=a'+b'\sqrt{-3}, \ a',b'\in \mathbb{Q}.
$$
Заменим $a'$ и $b'$ на ближайшие к ним целые числа $a$ и $b$. Пусть $\lambda=a+b\sqrt{-3},\ \lambda' -\lambda=\epsilon$. Тогда

$$
\alpha-\lambda\beta=\alpha-\lambda'\beta+\epsilon\beta=\epsilon\beta.
$$
$$
N(\alpha-\lambda\beta)=N(\epsilon\beta)=N(\epsilon)N(\beta),
$$

$$
N(\epsilon)=N(\lambda'-\lambda)=(a'-a)^2+3(b'-b)^2\leq (\frac{1}{2})^2+3(\frac{1}{2})^2=1.
$$

Если бы не существовало таких $\alpha,\beta\in R$, для которых $N(\epsilon)=1$, то $R$ было бы евклидовым кольцом. Следовательно, такие $\alpha,\beta$ существуют.
Если выполнено
$$
N(\alpha-\lambda\beta)=N(\epsilon\beta)=N(\epsilon)N(\beta), \ \ N(\epsilon)=1,
$$
то изменение $a$ и $b$ не может привести к уменьшению нормы остатка от деления:
$(a'-a)^2+3(b'-b)^2$ может только увеличиться.

Теперь нашей ближайшей задачей будет найти такие
ненулевые элементы $\alpha,\beta$ в $R$, для которых норма остатка от деления (определённого выше) $\alpha$ на $\beta$ равна норме делителя:
$$
\alpha = \beta\lambda+r, \ N(r)=N(\beta).
$$

Пусть $\alpha = a+b\sqrt{-3}, \ \beta=c+d\sqrt{-3}$.
$$
\lambda'=\beta^{-1}\alpha=\frac{c-d\sqrt{-3}}{c^2+3d^2}(a+b\sqrt{-3})=\frac{ac+3bd}{c^2+3d^2}+\frac{bc-ad}{c^2+3d^2}\sqrt{-3}.
$$

Будем искать такое частное $\lambda = l+m\sqrt{-3},\ l,m\in\mathbb{Z}$, что
$$
\frac{ac+3bd}{c^2+3d^2}=l+\frac{1}{2},\ \ \frac{bc-ad}{c^2+3d^2}=m+\frac{1}{2}.
$$

Если мы найдём целочисленные решения этих уравнений, нам не нужно будет проверять, отличен ли остаток $r$ от $\beta$. Действительно, если бы было $r=\beta$, то
$$
\alpha=\beta\lambda+\beta=\beta(\lambda+1).
$$
В последнем выражении вещественная компонента $\lambda$ отличается от вещественной компоненты $\lambda'$ на $1$, а
в наших уравнениях --- на $\frac{1}{2}$.

Ниже представлен PHP-скрипт, в котором решения (не все, конечно, только небольшое количество конкретных примеров) ищутся простым перебором.

<?php

class Element
{
    public int $a = 0;
    public int $b = 0;

    function __construct(int $a, int $b)
    {
        $this->a = $a;
        $this->b = $b;
    }

    function multiply(Element $e): Element
    {
        return new Element($this->a * $e->a - 3 * $this->b * $e->b, $this->a * $e->b + $this->b * $e->a);
    }

    function add(Element $e): Element
    {
        return new Element($this->a + $e->a, $this->b + $e->b);
    }

    function subtract(Element $e): Element
    {
        return new Element($this->a - $e->a, $this->b - $e->b);
    }

    function getNorm(): int
    {
        return $this->a ** 2 + 3 * $this->b ** 2;
    }

    function __toString(): string
    {
        $sign = $this->b <= 0 ? '' : '+';
        $imaginary = $this->b === 0 ? '': "{$this->b}\sqrt{-3}";

        if ($this->a === 0 && $this->b !== 0) {
            return $imaginary;
        } else {
            return "{$this->a}{$sign}{$imaginary}";
        }
    }

    function equals(Element $e)
    {
        return $this->a === $e->a && $this->b === $e->b;
    }
}

$N = 5;

echo "<ol>\n";

for ($a = -$N; $a <= $N; $a++) {
    for ($b = -$N; $b <= $N; $b++) {
        if ($a === 0 && $b === 0) {
            continue;
        }
        for ($c = -$N; $c <= $N; $c++) {
            for ($d = -$N; $d <= $N; $d++) {
                if ($c === 0 && $d === 0) {
                    continue;
                }
                for ($l = -$N; $l <= $N; $l++) {
                    for ($m = -$N; $m <= $N; $m++) {
                        if ($l === 0 && $m === 0) {
                            continue;
                        }

                        $s = $c ** 2 + 3 * $d ** 2;
                        if (2*($a*$c+3*$b*$d)-(2*$l+1)*$s === 0 &&
                            2*($b*$c-$a*$d)-(2*$m+1)*$s === 0) {

                            $alpha  = new Element($a, $b);
                            $beta   = new Element($c, $d);
                            $lambda = new Element($l, $m);

                            $r = $alpha->subtract( $beta->multiply($lambda));

                            echo "<li>\n";
                            echo "$$ $alpha = ($beta) \\cdot ($lambda) + ($r),$$\n";
                            echo "$$ N($beta) = N($r).$$\n";
                            echo "</li>\n";
                        }
                    }
                }
            }
        }
    }
}
echo "</ol>\n";

Вывод скрипта:

1. $$ -5-5\sqrt{-3} = (-5+5\sqrt{-3})(-1) + (-10),$$
   $$ N(-5+5\sqrt{-3}) = N(-10).$$
2. $$ -5-5\sqrt{-3} = (-2)(2+2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
   $$ N(-2) = N(-1-1\sqrt{-3}).$$
3. $$ -5-5\sqrt{-3} = (-1+1\sqrt{-3})(-3+2\sqrt{-3}) + (-2),$$
   $$ N(-1+1\sqrt{-3}) = N(-2).$$
4. $$ -5-5\sqrt{-3} = (1-1\sqrt{-3})(2-3\sqrt{-3}) + (2),$$
   $$ N(1-1\sqrt{-3}) = N(2).$$
5. $$ -5-5\sqrt{-3} = (2)(-3-3\sqrt{-3}) + (1+1\sqrt{-3}),$$
   $$ N(2) = N(1+1\sqrt{-3}).$$
6. $$ -5-5\sqrt{-3} = (5-5\sqrt{-3})(-1\sqrt{-3}) + (10),$$
   $$ N(5-5\sqrt{-3}) = N(10).$$
7. $$ -5-3\sqrt{-3} = (-2)(2+1\sqrt{-3}) + (-1-1\sqrt{-3}),$$
   $$ N(-2) = N(-1-1\sqrt{-3}).$$
8. $$ -5-3\sqrt{-3} = (-2+4\sqrt{-3})(-1) + (-7+1\sqrt{-3}),$$
   $$ N(-2+4\sqrt{-3}) = N(-7+1\sqrt{-3}).$$
9. $$ -5-3\sqrt{-3} = (-1-1\sqrt{-3})(3-1\sqrt{-3}) + (1-1\sqrt{-3}),$$
   $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
10. $$ -5-3\sqrt{-3} = (1+1\sqrt{-3})(-4) + (-1+1\sqrt{-3}),$$
    $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
11. $$ -5-3\sqrt{-3} = (2-4\sqrt{-3})(-1\sqrt{-3}) + (7-1\sqrt{-3}),$$
    $$ N(2-4\sqrt{-3}) = N(7-1\sqrt{-3}).$$
12. $$ -5-3\sqrt{-3} = (2)(-3-2\sqrt{-3}) + (1+1\sqrt{-3}),$$
    $$ N(2) = N(1+1\sqrt{-3}).$$
13. $$ -5-1\sqrt{-3} = (-2)(2) + (-1-1\sqrt{-3}),$$
    $$ N(-2) = N(-1-1\sqrt{-3}).$$
14. $$ -5-1\sqrt{-3} = (-1-3\sqrt{-3})(-1\sqrt{-3}) + (4-2\sqrt{-3}),$$
    $$ N(-1-3\sqrt{-3}) = N(4-2\sqrt{-3}).$$
15. $$ -5-1\sqrt{-3} = (-1+1\sqrt{-3})(1\sqrt{-3}) + (-2),$$
    $$ N(-1+1\sqrt{-3}) = N(-2).$$
16. $$ -5-1\sqrt{-3} = (1-1\sqrt{-3})(-1-2\sqrt{-3}) + (2),$$
    $$ N(1-1\sqrt{-3}) = N(2).$$
17. $$ -5-1\sqrt{-3} = (1+3\sqrt{-3})(-1) + (-4+2\sqrt{-3}),$$
    $$ N(1+3\sqrt{-3}) = N(-4+2\sqrt{-3}).$$
18. $$ -5-1\sqrt{-3} = (2)(-3-1\sqrt{-3}) + (1+1\sqrt{-3}),$$
    $$ N(2) = N(1+1\sqrt{-3}).$$
19. $$ -5-1\sqrt{-3} = (4-2\sqrt{-3})(-1-1\sqrt{-3}) + (5+1\sqrt{-3}),$$
    $$ N(4-2\sqrt{-3}) = N(5+1\sqrt{-3}).$$
20. $$ -5+1\sqrt{-3} = (-4-2\sqrt{-3})(-1\sqrt{-3}) + (1-3\sqrt{-3}),$$
    $$ N(-4-2\sqrt{-3}) = N(1-3\sqrt{-3}).$$
21. $$ -5+1\sqrt{-3} = (-2)(2-1\sqrt{-3}) + (-1-1\sqrt{-3}),$$
    $$ N(-2) = N(-1-1\sqrt{-3}).$$
22. $$ -5+1\sqrt{-3} = (-1-1\sqrt{-3})(-2\sqrt{-3}) + (1-1\sqrt{-3}),$$
    $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
23. $$ -5+1\sqrt{-3} = (1-3\sqrt{-3})(-1-1\sqrt{-3}) + (5-1\sqrt{-3}),$$
    $$ N(1-3\sqrt{-3}) = N(5-1\sqrt{-3}).$$
24. $$ -5+1\sqrt{-3} = (1+1\sqrt{-3})(-1+1\sqrt{-3}) + (-1+1\sqrt{-3}),$$
    $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
25. $$ -5+1\sqrt{-3} = (2)(-3) + (1+1\sqrt{-3}),$$
    $$ N(2) = N(1+1\sqrt{-3}).$$
26. $$ -5+1\sqrt{-3} = (4+2\sqrt{-3})(-1) + (-1+3\sqrt{-3}),$$
    $$ N(4+2\sqrt{-3}) = N(-1+3\sqrt{-3}).$$
27. $$ -5+3\sqrt{-3} = (-2-4\sqrt{-3})(-1-1\sqrt{-3}) + (5-3\sqrt{-3}),$$
    $$ N(-2-4\sqrt{-3}) = N(5-3\sqrt{-3}).$$
28. $$ -5+3\sqrt{-3} = (-2)(2-2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
    $$ N(-2) = N(-1-1\sqrt{-3}).$$
29. $$ -5+3\sqrt{-3} = (-1+1\sqrt{-3})(3) + (-2),$$
    $$ N(-1+1\sqrt{-3}) = N(-2).$$
30. $$ -5+3\sqrt{-3} = (1-1\sqrt{-3})(-4-1\sqrt{-3}) + (2),$$
    $$ N(1-1\sqrt{-3}) = N(2).$$
31. $$ -5+3\sqrt{-3} = (2)(-3+1\sqrt{-3}) + (1+1\sqrt{-3}),$$
    $$ N(2) = N(1+1\sqrt{-3}).$$
32. $$ -5+5\sqrt{-3} = (-5-5\sqrt{-3})(-1-1\sqrt{-3}) + (5-5\sqrt{-3}),$$
    $$ N(-5-5\sqrt{-3}) = N(5-5\sqrt{-3}).$$
33. $$ -5+5\sqrt{-3} = (-2)(2-3\sqrt{-3}) + (-1-1\sqrt{-3}),$$
    $$ N(-2) = N(-1-1\sqrt{-3}).$$
34. $$ -5+5\sqrt{-3} = (-1-1\sqrt{-3})(-3-3\sqrt{-3}) + (1-1\sqrt{-3}),$$
    $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
35. $$ -5+5\sqrt{-3} = (1+1\sqrt{-3})(2+2\sqrt{-3}) + (-1+1\sqrt{-3}),$$
    $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
36. $$ -5+5\sqrt{-3} = (2)(-3+2\sqrt{-3}) + (1+1\sqrt{-3}),$$
    $$ N(2) = N(1+1\sqrt{-3}).$$
37. $$ -4-4\sqrt{-3} = (-4+4\sqrt{-3})(-1) + (-8),$$
    $$ N(-4+4\sqrt{-3}) = N(-8).$$
38. $$ -4-4\sqrt{-3} = (4-4\sqrt{-3})(-1\sqrt{-3}) + (8),$$
    $$ N(4-4\sqrt{-3}) = N(8).$$
39. $$ -4-2\sqrt{-3} = (-1-1\sqrt{-3})(2-1\sqrt{-3}) + (1-1\sqrt{-3}),$$
    $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
40. $$ -4-2\sqrt{-3} = (-1+1\sqrt{-3})(-1+1\sqrt{-3}) + (-2),$$
    $$ N(-1+1\sqrt{-3}) = N(-2).$$
41. $$ -4-2\sqrt{-3} = (-1+3\sqrt{-3})(-1) + (-5+1\sqrt{-3}),$$
    $$ N(-1+3\sqrt{-3}) = N(-5+1\sqrt{-3}).$$
42. $$ -4-2\sqrt{-3} = (1-3\sqrt{-3})(-1\sqrt{-3}) + (5-1\sqrt{-3}),$$
    $$ N(1-3\sqrt{-3}) = N(5-1\sqrt{-3}).$$
43. $$ -4-2\sqrt{-3} = (1-1\sqrt{-3})(-2\sqrt{-3}) + (2),$$
    $$ N(1-1\sqrt{-3}) = N(2).$$
44. $$ -4-2\sqrt{-3} = (1+1\sqrt{-3})(-3) + (-1+1\sqrt{-3}),$$
    $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
45. $$ -4-2\sqrt{-3} = (5-1\sqrt{-3})(-1-1\sqrt{-3}) + (4+2\sqrt{-3}),$$
    $$ N(5-1\sqrt{-3}) = N(4+2\sqrt{-3}).$$
46. $$ -4 = (-2-2\sqrt{-3})(-1\sqrt{-3}) + (2-2\sqrt{-3}),$$
    $$ N(-2-2\sqrt{-3}) = N(2-2\sqrt{-3}).$$
47. $$ -4 = (2-2\sqrt{-3})(-1-1\sqrt{-3}) + (4),$$
    $$ N(2-2\sqrt{-3}) = N(4).$$
48. $$ -4 = (2+2\sqrt{-3})(-1) + (-2+2\sqrt{-3}),$$
    $$ N(2+2\sqrt{-3}) = N(-2+2\sqrt{-3}).$$
49. $$ -4+2\sqrt{-3} = (-5-1\sqrt{-3})(-1\sqrt{-3}) + (-1-3\sqrt{-3}),$$
    $$ N(-5-1\sqrt{-3}) = N(-1-3\sqrt{-3}).$$
50. $$ -4+2\sqrt{-3} = (-1-3\sqrt{-3})(-1-1\sqrt{-3}) + (4-2\sqrt{-3}),$$
    $$ N(-1-3\sqrt{-3}) = N(4-2\sqrt{-3}).$$
51. $$ -4+2\sqrt{-3} = (-1-1\sqrt{-3})(-1-2\sqrt{-3}) + (1-1\sqrt{-3}),$$
    $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
52. $$ -4+2\sqrt{-3} = (-1+1\sqrt{-3})(2) + (-2),$$
    $$ N(-1+1\sqrt{-3}) = N(-2).$$
53. $$ -4+2\sqrt{-3} = (1-1\sqrt{-3})(-3-1\sqrt{-3}) + (2),$$
    $$ N(1-1\sqrt{-3}) = N(2).$$
54. $$ -4+2\sqrt{-3} = (1+1\sqrt{-3})(1\sqrt{-3}) + (-1+1\sqrt{-3}),$$
    $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
55. $$ -4+2\sqrt{-3} = (5+1\sqrt{-3})(-1) + (1+3\sqrt{-3}),$$
    $$ N(5+1\sqrt{-3}) = N(1+3\sqrt{-3}).$$
56. $$ -4+4\sqrt{-3} = (-4-4\sqrt{-3})(-1-1\sqrt{-3}) + (4-4\sqrt{-3}),$$
    $$ N(-4-4\sqrt{-3}) = N(4-4\sqrt{-3}).$$
57. $$ -3-5\sqrt{-3} = (-4-2\sqrt{-3})(1) + (1-3\sqrt{-3}),$$
    $$ N(-4-2\sqrt{-3}) = N(1-3\sqrt{-3}).$$
58. $$ -3-5\sqrt{-3} = (-3+1\sqrt{-3})(-1+1\sqrt{-3}) + (-3-1\sqrt{-3}),$$
    $$ N(-3+1\sqrt{-3}) = N(-3-1\sqrt{-3}).$$
59. $$ -3-5\sqrt{-3} = (-2)(1+2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
    $$ N(-2) = N(-1-1\sqrt{-3}).$$
60. $$ -3-5\sqrt{-3} = (-1-1\sqrt{-3})(4) + (1-1\sqrt{-3}),$$
    $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
61. $$ -3-5\sqrt{-3} = (-1+3\sqrt{-3})(-2) + (-5+1\sqrt{-3}),$$
    $$ N(-1+3\sqrt{-3}) = N(-5+1\sqrt{-3}).$$
62. $$ -3-5\sqrt{-3} = (-2\sqrt{-3})(2-1\sqrt{-3}) + (3-1\sqrt{-3}),$$
    $$ N(-2\sqrt{-3}) = N(3-1\sqrt{-3}).$$
63. $$ -3-5\sqrt{-3} = (2\sqrt{-3})(-3) + (-3+1\sqrt{-3}),$$
    $$ N(2\sqrt{-3}) = N(-3+1\sqrt{-3}).$$
64. $$ -3-5\sqrt{-3} = (1-3\sqrt{-3})(1-1\sqrt{-3}) + (5-1\sqrt{-3}),$$
    $$ N(1-3\sqrt{-3}) = N(5-1\sqrt{-3}).$$
65. $$ -3-5\sqrt{-3} = (1+1\sqrt{-3})(-5-1\sqrt{-3}) + (-1+1\sqrt{-3}),$$
    $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
66. $$ -3-5\sqrt{-3} = (2)(-2-3\sqrt{-3}) + (1+1\sqrt{-3}),$$
    $$ N(2) = N(1+1\sqrt{-3}).$$
67. $$ -3-5\sqrt{-3} = (3-1\sqrt{-3})(-2\sqrt{-3}) + (3+1\sqrt{-3}),$$
    $$ N(3-1\sqrt{-3}) = N(3+1\sqrt{-3}).$$
68. $$ -3-5\sqrt{-3} = (4+2\sqrt{-3})(-2-1\sqrt{-3}) + (-1+3\sqrt{-3}),$$
    $$ N(4+2\sqrt{-3}) = N(-1+3\sqrt{-3}).$$
69. $$ -3-3\sqrt{-3} = (-3-1\sqrt{-3})(1) + (-2\sqrt{-3}),$$
    $$ N(-3-1\sqrt{-3}) = N(-2\sqrt{-3}).$$
70. $$ -3-3\sqrt{-3} = (-3+3\sqrt{-3})(-1) + (-6),$$
    $$ N(-3+3\sqrt{-3}) = N(-6).$$
71. $$ -3-3\sqrt{-3} = (-2)(1+1\sqrt{-3}) + (-1-1\sqrt{-3}),$$
    $$ N(-2) = N(-1-1\sqrt{-3}).$$
72. $$ -3-3\sqrt{-3} = (-1+1\sqrt{-3})(-2+1\sqrt{-3}) + (-2),$$
    $$ N(-1+1\sqrt{-3}) = N(-2).$$
73. $$ -3-3\sqrt{-3} = (-2\sqrt{-3})(1-1\sqrt{-3}) + (3-1\sqrt{-3}),$$
    $$ N(-2\sqrt{-3}) = N(3-1\sqrt{-3}).$$
74. $$ -3-3\sqrt{-3} = (2\sqrt{-3})(-2) + (-3+1\sqrt{-3}),$$
    $$ N(2\sqrt{-3}) = N(-3+1\sqrt{-3}).$$
75. $$ -3-3\sqrt{-3} = (1-1\sqrt{-3})(1-2\sqrt{-3}) + (2),$$
    $$ N(1-1\sqrt{-3}) = N(2).$$
76. $$ -3-3\sqrt{-3} = (2)(-2-2\sqrt{-3}) + (1+1\sqrt{-3}),$$
    $$ N(2) = N(1+1\sqrt{-3}).$$
77. $$ -3-3\sqrt{-3} = (3-3\sqrt{-3})(-1\sqrt{-3}) + (6),$$
    $$ N(3-3\sqrt{-3}) = N(6).$$
78. $$ -3-3\sqrt{-3} = (3+1\sqrt{-3})(-2-1\sqrt{-3}) + (2\sqrt{-3}),$$
    $$ N(3+1\sqrt{-3}) = N(2\sqrt{-3}).$$
79. $$ -3-1\sqrt{-3} = (-2)(1) + (-1-1\sqrt{-3}),$$
    $$ N(-2) = N(-1-1\sqrt{-3}).$$
80. $$ -3-1\sqrt{-3} = (-1-1\sqrt{-3})(1-1\sqrt{-3}) + (1-1\sqrt{-3}),$$
    $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
81. $$ -3-1\sqrt{-3} = (-2\sqrt{-3})(-1\sqrt{-3}) + (3-1\sqrt{-3}),$$
    $$ N(-2\sqrt{-3}) = N(3-1\sqrt{-3}).$$
82. $$ -3-1\sqrt{-3} = (2\sqrt{-3})(-1) + (-3+1\sqrt{-3}),$$
    $$ N(2\sqrt{-3}) = N(-3+1\sqrt{-3}).$$
83. $$ -3-1\sqrt{-3} = (1+1\sqrt{-3})(-2) + (-1+1\sqrt{-3}),$$
    $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
84. $$ -3-1\sqrt{-3} = (2)(-2-1\sqrt{-3}) + (1+1\sqrt{-3}),$$
    $$ N(2) = N(1+1\sqrt{-3}).$$
85. $$ -3-1\sqrt{-3} = (3-1\sqrt{-3})(-1-1\sqrt{-3}) + (3+1\sqrt{-3}),$$
    $$ N(3-1\sqrt{-3}) = N(3+1\sqrt{-3}).$$
86. $$ -3+1\sqrt{-3} = (-3-1\sqrt{-3})(-1\sqrt{-3}) + (-2\sqrt{-3}),$$
    $$ N(-3-1\sqrt{-3}) = N(-2\sqrt{-3}).$$
87. $$ -3+1\sqrt{-3} = (-2)(1-1\sqrt{-3}) + (-1-1\sqrt{-3}),$$
    $$ N(-2) = N(-1-1\sqrt{-3}).$$
88. $$ -3+1\sqrt{-3} = (-1+1\sqrt{-3})(1) + (-2),$$
    $$ N(-1+1\sqrt{-3}) = N(-2).$$
89. $$ -3+1\sqrt{-3} = (-2\sqrt{-3})(-1-1\sqrt{-3}) + (3-1\sqrt{-3}),$$
    $$ N(-2\sqrt{-3}) = N(3-1\sqrt{-3}).$$
90. $$ -3+1\sqrt{-3} = (1-1\sqrt{-3})(-2-1\sqrt{-3}) + (2),$$
    $$ N(1-1\sqrt{-3}) = N(2).$$
91. $$ -3+1\sqrt{-3} = (2)(-2) + (1+1\sqrt{-3}),$$
    $$ N(2) = N(1+1\sqrt{-3}).$$
92. $$ -3+1\sqrt{-3} = (3+1\sqrt{-3})(-1) + (2\sqrt{-3}),$$
    $$ N(3+1\sqrt{-3}) = N(2\sqrt{-3}).$$
93. $$ -3+3\sqrt{-3} = (-3-3\sqrt{-3})(-1-1\sqrt{-3}) + (3-3\sqrt{-3}),$$
    $$ N(-3-3\sqrt{-3}) = N(3-3\sqrt{-3}).$$
94. $$ -3+3\sqrt{-3} = (-3+1\sqrt{-3})(1-1\sqrt{-3}) + (-3-1\sqrt{-3}),$$
    $$ N(-3+1\sqrt{-3}) = N(-3-1\sqrt{-3}).$$
95. $$ -3+3\sqrt{-3} = (-2)(1-2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
    $$ N(-2) = N(-1-1\sqrt{-3}).$$
96. $$ -3+3\sqrt{-3} = (-1-1\sqrt{-3})(-2-2\sqrt{-3}) + (1-1\sqrt{-3}),$$
    $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
97. $$ -3+3\sqrt{-3} = (-2\sqrt{-3})(-2-1\sqrt{-3}) + (3-1\sqrt{-3}),$$
    $$ N(-2\sqrt{-3}) = N(3-1\sqrt{-3}).$$
98. $$ -3+3\sqrt{-3} = (2\sqrt{-3})(1) + (-3+1\sqrt{-3}),$$
    $$ N(2\sqrt{-3}) = N(-3+1\sqrt{-3}).$$
99. $$ -3+3\sqrt{-3} = (1+1\sqrt{-3})(1+1\sqrt{-3}) + (-1+1\sqrt{-3}),$$
    $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
100. $$ -3+3\sqrt{-3} = (2)(-2+1\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
101. $$ -3+3\sqrt{-3} = (3-1\sqrt{-3})(-2) + (3+1\sqrt{-3}),$$
     $$ N(3-1\sqrt{-3}) = N(3+1\sqrt{-3}).$$
102. $$ -3+5\sqrt{-3} = (-4+2\sqrt{-3})(1-1\sqrt{-3}) + (-5-1\sqrt{-3}),$$
     $$ N(-4+2\sqrt{-3}) = N(-5-1\sqrt{-3}).$$
103. $$ -3+5\sqrt{-3} = (-3-1\sqrt{-3})(-1-2\sqrt{-3}) + (-2\sqrt{-3}),$$
     $$ N(-3-1\sqrt{-3}) = N(-2\sqrt{-3}).$$
104. $$ -3+5\sqrt{-3} = (-2)(1-3\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
105. $$ -3+5\sqrt{-3} = (-1-3\sqrt{-3})(-2-1\sqrt{-3}) + (4-2\sqrt{-3}),$$
     $$ N(-1-3\sqrt{-3}) = N(4-2\sqrt{-3}).$$
106. $$ -3+5\sqrt{-3} = (-1+1\sqrt{-3})(4-1\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
107. $$ -3+5\sqrt{-3} = (-2\sqrt{-3})(-3-1\sqrt{-3}) + (3-1\sqrt{-3}),$$
     $$ N(-2\sqrt{-3}) = N(3-1\sqrt{-3}).$$
108. $$ -3+5\sqrt{-3} = (2\sqrt{-3})(2) + (-3+1\sqrt{-3}),$$
     $$ N(2\sqrt{-3}) = N(-3+1\sqrt{-3}).$$
109. $$ -3+5\sqrt{-3} = (1-1\sqrt{-3})(-5) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
110. $$ -3+5\sqrt{-3} = (1+3\sqrt{-3})(1) + (-4+2\sqrt{-3}),$$
     $$ N(1+3\sqrt{-3}) = N(-4+2\sqrt{-3}).$$
111. $$ -3+5\sqrt{-3} = (2)(-2+2\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
112. $$ -3+5\sqrt{-3} = (3+1\sqrt{-3})(1\sqrt{-3}) + (2\sqrt{-3}),$$
     $$ N(3+1\sqrt{-3}) = N(2\sqrt{-3}).$$
113. $$ -3+5\sqrt{-3} = (4-2\sqrt{-3})(-2) + (5+1\sqrt{-3}),$$
     $$ N(4-2\sqrt{-3}) = N(5+1\sqrt{-3}).$$
114. $$ -2-4\sqrt{-3} = (-5+3\sqrt{-3})(-1) + (-7-1\sqrt{-3}),$$
     $$ N(-5+3\sqrt{-3}) = N(-7-1\sqrt{-3}).$$
115. $$ -2-4\sqrt{-3} = (-1-1\sqrt{-3})(3) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
116. $$ -2-4\sqrt{-3} = (-1+1\sqrt{-3})(-3+1\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
117. $$ -2-4\sqrt{-3} = (1-1\sqrt{-3})(2-2\sqrt{-3}) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
118. $$ -2-4\sqrt{-3} = (1+1\sqrt{-3})(-4-1\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
119. $$ -2-4\sqrt{-3} = (5-3\sqrt{-3})(-1\sqrt{-3}) + (7+1\sqrt{-3}),$$
     $$ N(5-3\sqrt{-3}) = N(7+1\sqrt{-3}).$$
120. $$ -2-2\sqrt{-3} = (-2+2\sqrt{-3})(-1) + (-4),$$
     $$ N(-2+2\sqrt{-3}) = N(-4).$$
121. $$ -2-2\sqrt{-3} = (2-2\sqrt{-3})(-1\sqrt{-3}) + (4),$$
     $$ N(2-2\sqrt{-3}) = N(4).$$
122. $$ -2-2\sqrt{-3} = (4)(-1-1\sqrt{-3}) + (2+2\sqrt{-3}),$$
     $$ N(4) = N(2+2\sqrt{-3}).$$
123. $$ -2 = (-1-1\sqrt{-3})(-1\sqrt{-3}) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
124. $$ -2 = (1-1\sqrt{-3})(-1-1\sqrt{-3}) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
125. $$ -2 = (1+1\sqrt{-3})(-1) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
126. $$ -2+2\sqrt{-3} = (-4)(-1\sqrt{-3}) + (-2-2\sqrt{-3}),$$
     $$ N(-4) = N(-2-2\sqrt{-3}).$$
127. $$ -2+2\sqrt{-3} = (-2-2\sqrt{-3})(-1-1\sqrt{-3}) + (2-2\sqrt{-3}),$$
     $$ N(-2-2\sqrt{-3}) = N(2-2\sqrt{-3}).$$
128. $$ -2+2\sqrt{-3} = (4)(-1) + (2+2\sqrt{-3}),$$
     $$ N(4) = N(2+2\sqrt{-3}).$$
129. $$ -2+4\sqrt{-3} = (-5-3\sqrt{-3})(-1-1\sqrt{-3}) + (2-4\sqrt{-3}),$$
     $$ N(-5-3\sqrt{-3}) = N(2-4\sqrt{-3}).$$
130. $$ -2+4\sqrt{-3} = (-1-1\sqrt{-3})(-3-2\sqrt{-3}) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
131. $$ -2+4\sqrt{-3} = (-1+1\sqrt{-3})(3-1\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
132. $$ -2+4\sqrt{-3} = (1-1\sqrt{-3})(-4) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
133. $$ -2+4\sqrt{-3} = (1+1\sqrt{-3})(2+1\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
134. $$ -1-5\sqrt{-3} = (-2)(2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
135. $$ -1-5\sqrt{-3} = (-1+1\sqrt{-3})(-4+1\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
136. $$ -1-5\sqrt{-3} = (1-1\sqrt{-3})(3-2\sqrt{-3}) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
137. $$ -1-5\sqrt{-3} = (2)(-1-3\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
138. $$ -1-3\sqrt{-3} = (-4+2\sqrt{-3})(-1) + (-5-1\sqrt{-3}),$$
     $$ N(-4+2\sqrt{-3}) = N(-5-1\sqrt{-3}).$$
139. $$ -1-3\sqrt{-3} = (-2)(1\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
140. $$ -1-3\sqrt{-3} = (-1-1\sqrt{-3})(2) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
141. $$ -1-3\sqrt{-3} = (1+1\sqrt{-3})(-3-1\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
142. $$ -1-3\sqrt{-3} = (2)(-1-2\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
143. $$ -1-3\sqrt{-3} = (4-2\sqrt{-3})(-1\sqrt{-3}) + (5+1\sqrt{-3}),$$
     $$ N(4-2\sqrt{-3}) = N(5+1\sqrt{-3}).$$
144. $$ -1-3\sqrt{-3} = (5+1\sqrt{-3})(-1-1\sqrt{-3}) + (1+3\sqrt{-3}),$$
     $$ N(5+1\sqrt{-3}) = N(1+3\sqrt{-3}).$$
145. $$ -1-1\sqrt{-3} = (-1+1\sqrt{-3})(-1) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
146. $$ -1-1\sqrt{-3} = (1-1\sqrt{-3})(-1\sqrt{-3}) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
147. $$ -1-1\sqrt{-3} = (2)(-1-1\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
148. $$ -1+1\sqrt{-3} = (-2)(-1\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
149. $$ -1+1\sqrt{-3} = (-1-1\sqrt{-3})(-1-1\sqrt{-3}) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
150. $$ -1+1\sqrt{-3} = (2)(-1) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
151. $$ -1+3\sqrt{-3} = (-5+1\sqrt{-3})(-1\sqrt{-3}) + (-4-2\sqrt{-3}),$$
     $$ N(-5+1\sqrt{-3}) = N(-4-2\sqrt{-3}).$$
152. $$ -1+3\sqrt{-3} = (-4-2\sqrt{-3})(-1-1\sqrt{-3}) + (1-3\sqrt{-3}),$$
     $$ N(-4-2\sqrt{-3}) = N(1-3\sqrt{-3}).$$
153. $$ -1+3\sqrt{-3} = (-2)(-2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
154. $$ -1+3\sqrt{-3} = (-1+1\sqrt{-3})(2-1\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
155. $$ -1+3\sqrt{-3} = (1-1\sqrt{-3})(-3) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
156. $$ -1+3\sqrt{-3} = (2)(-1+1\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
157. $$ -1+3\sqrt{-3} = (5-1\sqrt{-3})(-1) + (4+2\sqrt{-3}),$$
     $$ N(5-1\sqrt{-3}) = N(4+2\sqrt{-3}).$$
158. $$ -1+5\sqrt{-3} = (-2)(-3\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
159. $$ -1+5\sqrt{-3} = (-1-1\sqrt{-3})(-4-2\sqrt{-3}) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
160. $$ -1+5\sqrt{-3} = (1+1\sqrt{-3})(3+1\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
161. $$ -1+5\sqrt{-3} = (2)(-1+2\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
162. $$ -4\sqrt{-3} = (-2-2\sqrt{-3})(1) + (2-2\sqrt{-3}),$$
     $$ N(-2-2\sqrt{-3}) = N(2-2\sqrt{-3}).$$
163. $$ -4\sqrt{-3} = (-2+2\sqrt{-3})(-2) + (-4),$$
     $$ N(-2+2\sqrt{-3}) = N(-4).$$
164. $$ -4\sqrt{-3} = (2-2\sqrt{-3})(1-1\sqrt{-3}) + (4),$$
     $$ N(2-2\sqrt{-3}) = N(4).$$
165. $$ -4\sqrt{-3} = (2+2\sqrt{-3})(-2-1\sqrt{-3}) + (-2+2\sqrt{-3}),$$
     $$ N(2+2\sqrt{-3}) = N(-2+2\sqrt{-3}).$$
166. $$ -2\sqrt{-3} = (-3+1\sqrt{-3})(-1) + (-3-1\sqrt{-3}),$$
     $$ N(-3+1\sqrt{-3}) = N(-3-1\sqrt{-3}).$$
167. $$ -2\sqrt{-3} = (-1-1\sqrt{-3})(1) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
168. $$ -2\sqrt{-3} = (-1+1\sqrt{-3})(-2) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
169. $$ -2\sqrt{-3} = (1-1\sqrt{-3})(1-1\sqrt{-3}) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
170. $$ -2\sqrt{-3} = (1+1\sqrt{-3})(-2-1\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
171. $$ -2\sqrt{-3} = (3-1\sqrt{-3})(-1\sqrt{-3}) + (3+1\sqrt{-3}),$$
     $$ N(3-1\sqrt{-3}) = N(3+1\sqrt{-3}).$$
172. $$ -2\sqrt{-3} = (3+1\sqrt{-3})(-1-1\sqrt{-3}) + (2\sqrt{-3}),$$
     $$ N(3+1\sqrt{-3}) = N(2\sqrt{-3}).$$
173. $$ 2\sqrt{-3} = (-3-1\sqrt{-3})(-1-1\sqrt{-3}) + (-2\sqrt{-3}),$$
     $$ N(-3-1\sqrt{-3}) = N(-2\sqrt{-3}).$$
174. $$ 2\sqrt{-3} = (-3+1\sqrt{-3})(-1\sqrt{-3}) + (-3-1\sqrt{-3}),$$
     $$ N(-3+1\sqrt{-3}) = N(-3-1\sqrt{-3}).$$
175. $$ 2\sqrt{-3} = (-1-1\sqrt{-3})(-2-1\sqrt{-3}) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
176. $$ 2\sqrt{-3} = (-1+1\sqrt{-3})(1-1\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
177. $$ 2\sqrt{-3} = (1-1\sqrt{-3})(-2) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
178. $$ 2\sqrt{-3} = (1+1\sqrt{-3})(1) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
179. $$ 2\sqrt{-3} = (3-1\sqrt{-3})(-1) + (3+1\sqrt{-3}),$$
     $$ N(3-1\sqrt{-3}) = N(3+1\sqrt{-3}).$$
180. $$ 4\sqrt{-3} = (-2-2\sqrt{-3})(-2-1\sqrt{-3}) + (2-2\sqrt{-3}),$$
     $$ N(-2-2\sqrt{-3}) = N(2-2\sqrt{-3}).$$
181. $$ 4\sqrt{-3} = (-2+2\sqrt{-3})(1-1\sqrt{-3}) + (-4),$$
     $$ N(-2+2\sqrt{-3}) = N(-4).$$
182. $$ 4\sqrt{-3} = (2-2\sqrt{-3})(-2) + (4),$$
     $$ N(2-2\sqrt{-3}) = N(4).$$
183. $$ 4\sqrt{-3} = (2+2\sqrt{-3})(1) + (-2+2\sqrt{-3}),$$
     $$ N(2+2\sqrt{-3}) = N(-2+2\sqrt{-3}).$$
184. $$ 1-5\sqrt{-3} = (-2)(-1+2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
185. $$ 1-5\sqrt{-3} = (-1-1\sqrt{-3})(3+1\sqrt{-3}) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
186. $$ 1-5\sqrt{-3} = (1+1\sqrt{-3})(-4-2\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
187. $$ 1-5\sqrt{-3} = (2)(-3\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
188. $$ 1-3\sqrt{-3} = (-5+1\sqrt{-3})(-1) + (-4-2\sqrt{-3}),$$
     $$ N(-5+1\sqrt{-3}) = N(-4-2\sqrt{-3}).$$
189. $$ 1-3\sqrt{-3} = (-2)(-1+1\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
190. $$ 1-3\sqrt{-3} = (-1+1\sqrt{-3})(-3) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
191. $$ 1-3\sqrt{-3} = (1-1\sqrt{-3})(2-1\sqrt{-3}) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
192. $$ 1-3\sqrt{-3} = (2)(-2\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
193. $$ 1-3\sqrt{-3} = (4+2\sqrt{-3})(-1-1\sqrt{-3}) + (-1+3\sqrt{-3}),$$
     $$ N(4+2\sqrt{-3}) = N(-1+3\sqrt{-3}).$$
194. $$ 1-3\sqrt{-3} = (5-1\sqrt{-3})(-1\sqrt{-3}) + (4+2\sqrt{-3}),$$
     $$ N(5-1\sqrt{-3}) = N(4+2\sqrt{-3}).$$
195. $$ 1-1\sqrt{-3} = (-2)(-1) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
196. $$ 1-1\sqrt{-3} = (1+1\sqrt{-3})(-1-1\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
197. $$ 1-1\sqrt{-3} = (2)(-1\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
198. $$ 1+1\sqrt{-3} = (-2)(-1-1\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
199. $$ 1+1\sqrt{-3} = (-1+1\sqrt{-3})(-1\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
200. $$ 1+1\sqrt{-3} = (1-1\sqrt{-3})(-1) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
201. $$ 1+3\sqrt{-3} = (-5-1\sqrt{-3})(-1-1\sqrt{-3}) + (-1-3\sqrt{-3}),$$
     $$ N(-5-1\sqrt{-3}) = N(-1-3\sqrt{-3}).$$
202. $$ 1+3\sqrt{-3} = (-4+2\sqrt{-3})(-1\sqrt{-3}) + (-5-1\sqrt{-3}),$$
     $$ N(-4+2\sqrt{-3}) = N(-5-1\sqrt{-3}).$$
203. $$ 1+3\sqrt{-3} = (-2)(-1-2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
204. $$ 1+3\sqrt{-3} = (-1-1\sqrt{-3})(-3-1\sqrt{-3}) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
205. $$ 1+3\sqrt{-3} = (1+1\sqrt{-3})(2) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
206. $$ 1+3\sqrt{-3} = (2)(1\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
207. $$ 1+3\sqrt{-3} = (4-2\sqrt{-3})(-1) + (5+1\sqrt{-3}),$$
     $$ N(4-2\sqrt{-3}) = N(5+1\sqrt{-3}).$$
208. $$ 1+5\sqrt{-3} = (-2)(-1-3\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
209. $$ 1+5\sqrt{-3} = (-1+1\sqrt{-3})(3-2\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
210. $$ 1+5\sqrt{-3} = (1-1\sqrt{-3})(-4+1\sqrt{-3}) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
211. $$ 1+5\sqrt{-3} = (2)(2\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
212. $$ 2-4\sqrt{-3} = (-1-1\sqrt{-3})(2+1\sqrt{-3}) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
213. $$ 2-4\sqrt{-3} = (-1+1\sqrt{-3})(-4) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
214. $$ 2-4\sqrt{-3} = (1-1\sqrt{-3})(3-1\sqrt{-3}) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
215. $$ 2-4\sqrt{-3} = (1+1\sqrt{-3})(-3-2\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
216. $$ 2-4\sqrt{-3} = (5+3\sqrt{-3})(-1-1\sqrt{-3}) + (-2+4\sqrt{-3}),$$
     $$ N(5+3\sqrt{-3}) = N(-2+4\sqrt{-3}).$$
217. $$ 2-2\sqrt{-3} = (-4)(-1) + (-2-2\sqrt{-3}),$$
     $$ N(-4) = N(-2-2\sqrt{-3}).$$
218. $$ 2-2\sqrt{-3} = (2+2\sqrt{-3})(-1-1\sqrt{-3}) + (-2+2\sqrt{-3}),$$
     $$ N(2+2\sqrt{-3}) = N(-2+2\sqrt{-3}).$$
219. $$ 2-2\sqrt{-3} = (4)(-1\sqrt{-3}) + (2+2\sqrt{-3}),$$
     $$ N(4) = N(2+2\sqrt{-3}).$$
220. $$ 2 = (-1-1\sqrt{-3})(-1) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
221. $$ 2 = (-1+1\sqrt{-3})(-1-1\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
222. $$ 2 = (1+1\sqrt{-3})(-1\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
223. $$ 2+2\sqrt{-3} = (-4)(-1-1\sqrt{-3}) + (-2-2\sqrt{-3}),$$
     $$ N(-4) = N(-2-2\sqrt{-3}).$$
224. $$ 2+2\sqrt{-3} = (-2+2\sqrt{-3})(-1\sqrt{-3}) + (-4),$$
     $$ N(-2+2\sqrt{-3}) = N(-4).$$
225. $$ 2+2\sqrt{-3} = (2-2\sqrt{-3})(-1) + (4),$$
     $$ N(2-2\sqrt{-3}) = N(4).$$
226. $$ 2+4\sqrt{-3} = (-5+3\sqrt{-3})(-1\sqrt{-3}) + (-7-1\sqrt{-3}),$$
     $$ N(-5+3\sqrt{-3}) = N(-7-1\sqrt{-3}).$$
227. $$ 2+4\sqrt{-3} = (-1-1\sqrt{-3})(-4-1\sqrt{-3}) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
228. $$ 2+4\sqrt{-3} = (-1+1\sqrt{-3})(2-2\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
229. $$ 2+4\sqrt{-3} = (1-1\sqrt{-3})(-3+1\sqrt{-3}) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
230. $$ 2+4\sqrt{-3} = (1+1\sqrt{-3})(3) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
231. $$ 2+4\sqrt{-3} = (5-3\sqrt{-3})(-1) + (7+1\sqrt{-3}),$$
     $$ N(5-3\sqrt{-3}) = N(7+1\sqrt{-3}).$$
232. $$ 3-5\sqrt{-3} = (-4+2\sqrt{-3})(-2) + (-5-1\sqrt{-3}),$$
     $$ N(-4+2\sqrt{-3}) = N(-5-1\sqrt{-3}).$$
233. $$ 3-5\sqrt{-3} = (-3-1\sqrt{-3})(1\sqrt{-3}) + (-2\sqrt{-3}),$$
     $$ N(-3-1\sqrt{-3}) = N(-2\sqrt{-3}).$$
234. $$ 3-5\sqrt{-3} = (-2)(-2+2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
235. $$ 3-5\sqrt{-3} = (-1-3\sqrt{-3})(1) + (4-2\sqrt{-3}),$$
     $$ N(-1-3\sqrt{-3}) = N(4-2\sqrt{-3}).$$
236. $$ 3-5\sqrt{-3} = (-1+1\sqrt{-3})(-5) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
237. $$ 3-5\sqrt{-3} = (-2\sqrt{-3})(2) + (3-1\sqrt{-3}),$$
     $$ N(-2\sqrt{-3}) = N(3-1\sqrt{-3}).$$
238. $$ 3-5\sqrt{-3} = (2\sqrt{-3})(-3-1\sqrt{-3}) + (-3+1\sqrt{-3}),$$
     $$ N(2\sqrt{-3}) = N(-3+1\sqrt{-3}).$$
239. $$ 3-5\sqrt{-3} = (1-1\sqrt{-3})(4-1\sqrt{-3}) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
240. $$ 3-5\sqrt{-3} = (1+3\sqrt{-3})(-2-1\sqrt{-3}) + (-4+2\sqrt{-3}),$$
     $$ N(1+3\sqrt{-3}) = N(-4+2\sqrt{-3}).$$
241. $$ 3-5\sqrt{-3} = (2)(1-3\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
242. $$ 3-5\sqrt{-3} = (3+1\sqrt{-3})(-1-2\sqrt{-3}) + (2\sqrt{-3}),$$
     $$ N(3+1\sqrt{-3}) = N(2\sqrt{-3}).$$
243. $$ 3-5\sqrt{-3} = (4-2\sqrt{-3})(1-1\sqrt{-3}) + (5+1\sqrt{-3}),$$
     $$ N(4-2\sqrt{-3}) = N(5+1\sqrt{-3}).$$
244. $$ 3-3\sqrt{-3} = (-3+1\sqrt{-3})(-2) + (-3-1\sqrt{-3}),$$
     $$ N(-3+1\sqrt{-3}) = N(-3-1\sqrt{-3}).$$
245. $$ 3-3\sqrt{-3} = (-2)(-2+1\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
246. $$ 3-3\sqrt{-3} = (-1-1\sqrt{-3})(1+1\sqrt{-3}) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
247. $$ 3-3\sqrt{-3} = (-2\sqrt{-3})(1) + (3-1\sqrt{-3}),$$
     $$ N(-2\sqrt{-3}) = N(3-1\sqrt{-3}).$$
248. $$ 3-3\sqrt{-3} = (2\sqrt{-3})(-2-1\sqrt{-3}) + (-3+1\sqrt{-3}),$$
     $$ N(2\sqrt{-3}) = N(-3+1\sqrt{-3}).$$
249. $$ 3-3\sqrt{-3} = (1+1\sqrt{-3})(-2-2\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
250. $$ 3-3\sqrt{-3} = (2)(1-2\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
251. $$ 3-3\sqrt{-3} = (3-1\sqrt{-3})(1-1\sqrt{-3}) + (3+1\sqrt{-3}),$$
     $$ N(3-1\sqrt{-3}) = N(3+1\sqrt{-3}).$$
252. $$ 3-3\sqrt{-3} = (3+3\sqrt{-3})(-1-1\sqrt{-3}) + (-3+3\sqrt{-3}),$$
     $$ N(3+3\sqrt{-3}) = N(-3+3\sqrt{-3}).$$
253. $$ 3-1\sqrt{-3} = (-3-1\sqrt{-3})(-1) + (-2\sqrt{-3}),$$
     $$ N(-3-1\sqrt{-3}) = N(-2\sqrt{-3}).$$
254. $$ 3-1\sqrt{-3} = (-2)(-2) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
255. $$ 3-1\sqrt{-3} = (-1+1\sqrt{-3})(-2-1\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
256. $$ 3-1\sqrt{-3} = (2\sqrt{-3})(-1-1\sqrt{-3}) + (-3+1\sqrt{-3}),$$
     $$ N(2\sqrt{-3}) = N(-3+1\sqrt{-3}).$$
257. $$ 3-1\sqrt{-3} = (1-1\sqrt{-3})(1) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
258. $$ 3-1\sqrt{-3} = (2)(1-1\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
259. $$ 3-1\sqrt{-3} = (3+1\sqrt{-3})(-1\sqrt{-3}) + (2\sqrt{-3}),$$
     $$ N(3+1\sqrt{-3}) = N(2\sqrt{-3}).$$
260. $$ 3+1\sqrt{-3} = (-3+1\sqrt{-3})(-1-1\sqrt{-3}) + (-3-1\sqrt{-3}),$$
     $$ N(-3+1\sqrt{-3}) = N(-3-1\sqrt{-3}).$$
261. $$ 3+1\sqrt{-3} = (-2)(-2-1\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
262. $$ 3+1\sqrt{-3} = (-1-1\sqrt{-3})(-2) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
263. $$ 3+1\sqrt{-3} = (-2\sqrt{-3})(-1) + (3-1\sqrt{-3}),$$
     $$ N(-2\sqrt{-3}) = N(3-1\sqrt{-3}).$$
264. $$ 3+1\sqrt{-3} = (2\sqrt{-3})(-1\sqrt{-3}) + (-3+1\sqrt{-3}),$$
     $$ N(2\sqrt{-3}) = N(-3+1\sqrt{-3}).$$
265. $$ 3+1\sqrt{-3} = (1+1\sqrt{-3})(1-1\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
266. $$ 3+1\sqrt{-3} = (2)(1) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
267. $$ 3+3\sqrt{-3} = (-3-1\sqrt{-3})(-2-1\sqrt{-3}) + (-2\sqrt{-3}),$$
     $$ N(-3-1\sqrt{-3}) = N(-2\sqrt{-3}).$$
268. $$ 3+3\sqrt{-3} = (-3+3\sqrt{-3})(-1\sqrt{-3}) + (-6),$$
     $$ N(-3+3\sqrt{-3}) = N(-6).$$
269. $$ 3+3\sqrt{-3} = (-2)(-2-2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
270. $$ 3+3\sqrt{-3} = (-1+1\sqrt{-3})(1-2\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
271. $$ 3+3\sqrt{-3} = (-2\sqrt{-3})(-2) + (3-1\sqrt{-3}),$$
     $$ N(-2\sqrt{-3}) = N(3-1\sqrt{-3}).$$
272. $$ 3+3\sqrt{-3} = (2\sqrt{-3})(1-1\sqrt{-3}) + (-3+1\sqrt{-3}),$$
     $$ N(2\sqrt{-3}) = N(-3+1\sqrt{-3}).$$
273. $$ 3+3\sqrt{-3} = (1-1\sqrt{-3})(-2+1\sqrt{-3}) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
274. $$ 3+3\sqrt{-3} = (2)(1+1\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
275. $$ 3+3\sqrt{-3} = (3-3\sqrt{-3})(-1) + (6),$$
     $$ N(3-3\sqrt{-3}) = N(6).$$
276. $$ 3+3\sqrt{-3} = (3+1\sqrt{-3})(1) + (2\sqrt{-3}),$$
     $$ N(3+1\sqrt{-3}) = N(2\sqrt{-3}).$$
277. $$ 3+5\sqrt{-3} = (-4-2\sqrt{-3})(-2-1\sqrt{-3}) + (1-3\sqrt{-3}),$$
     $$ N(-4-2\sqrt{-3}) = N(1-3\sqrt{-3}).$$
278. $$ 3+5\sqrt{-3} = (-3+1\sqrt{-3})(-2\sqrt{-3}) + (-3-1\sqrt{-3}),$$
     $$ N(-3+1\sqrt{-3}) = N(-3-1\sqrt{-3}).$$
279. $$ 3+5\sqrt{-3} = (-2)(-2-3\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
280. $$ 3+5\sqrt{-3} = (-1-1\sqrt{-3})(-5-1\sqrt{-3}) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
281. $$ 3+5\sqrt{-3} = (-1+3\sqrt{-3})(1-1\sqrt{-3}) + (-5+1\sqrt{-3}),$$
     $$ N(-1+3\sqrt{-3}) = N(-5+1\sqrt{-3}).$$
282. $$ 3+5\sqrt{-3} = (-2\sqrt{-3})(-3) + (3-1\sqrt{-3}),$$
     $$ N(-2\sqrt{-3}) = N(3-1\sqrt{-3}).$$
283. $$ 3+5\sqrt{-3} = (2\sqrt{-3})(2-1\sqrt{-3}) + (-3+1\sqrt{-3}),$$
     $$ N(2\sqrt{-3}) = N(-3+1\sqrt{-3}).$$
284. $$ 3+5\sqrt{-3} = (1-3\sqrt{-3})(-2) + (5-1\sqrt{-3}),$$
     $$ N(1-3\sqrt{-3}) = N(5-1\sqrt{-3}).$$
285. $$ 3+5\sqrt{-3} = (1+1\sqrt{-3})(4) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
286. $$ 3+5\sqrt{-3} = (2)(1+2\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
287. $$ 3+5\sqrt{-3} = (3-1\sqrt{-3})(-1+1\sqrt{-3}) + (3+1\sqrt{-3}),$$
     $$ N(3-1\sqrt{-3}) = N(3+1\sqrt{-3}).$$
288. $$ 3+5\sqrt{-3} = (4+2\sqrt{-3})(1) + (-1+3\sqrt{-3}),$$
     $$ N(4+2\sqrt{-3}) = N(-1+3\sqrt{-3}).$$
289. $$ 4-4\sqrt{-3} = (4+4\sqrt{-3})(-1-1\sqrt{-3}) + (-4+4\sqrt{-3}),$$
     $$ N(4+4\sqrt{-3}) = N(-4+4\sqrt{-3}).$$
290. $$ 4-2\sqrt{-3} = (-5-1\sqrt{-3})(-1) + (-1-3\sqrt{-3}),$$
     $$ N(-5-1\sqrt{-3}) = N(-1-3\sqrt{-3}).$$
291. $$ 4-2\sqrt{-3} = (-1-1\sqrt{-3})(1\sqrt{-3}) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
292. $$ 4-2\sqrt{-3} = (-1+1\sqrt{-3})(-3-1\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
293. $$ 4-2\sqrt{-3} = (1-1\sqrt{-3})(2) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
294. $$ 4-2\sqrt{-3} = (1+1\sqrt{-3})(-1-2\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
295. $$ 4-2\sqrt{-3} = (1+3\sqrt{-3})(-1-1\sqrt{-3}) + (-4+2\sqrt{-3}),$$
     $$ N(1+3\sqrt{-3}) = N(-4+2\sqrt{-3}).$$
296. $$ 4-2\sqrt{-3} = (5+1\sqrt{-3})(-1\sqrt{-3}) + (1+3\sqrt{-3}),$$
     $$ N(5+1\sqrt{-3}) = N(1+3\sqrt{-3}).$$
297. $$ 4 = (-2-2\sqrt{-3})(-1) + (2-2\sqrt{-3}),$$
     $$ N(-2-2\sqrt{-3}) = N(2-2\sqrt{-3}).$$
298. $$ 4 = (-2+2\sqrt{-3})(-1-1\sqrt{-3}) + (-4),$$
     $$ N(-2+2\sqrt{-3}) = N(-4).$$
299. $$ 4 = (2+2\sqrt{-3})(-1\sqrt{-3}) + (-2+2\sqrt{-3}),$$
     $$ N(2+2\sqrt{-3}) = N(-2+2\sqrt{-3}).$$
300. $$ 4+2\sqrt{-3} = (-5+1\sqrt{-3})(-1-1\sqrt{-3}) + (-4-2\sqrt{-3}),$$
     $$ N(-5+1\sqrt{-3}) = N(-4-2\sqrt{-3}).$$
301. $$ 4+2\sqrt{-3} = (-1-1\sqrt{-3})(-3) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
302. $$ 4+2\sqrt{-3} = (-1+1\sqrt{-3})(-2\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
303. $$ 4+2\sqrt{-3} = (-1+3\sqrt{-3})(-1\sqrt{-3}) + (-5+1\sqrt{-3}),$$
     $$ N(-1+3\sqrt{-3}) = N(-5+1\sqrt{-3}).$$
304. $$ 4+2\sqrt{-3} = (1-3\sqrt{-3})(-1) + (5-1\sqrt{-3}),$$
     $$ N(1-3\sqrt{-3}) = N(5-1\sqrt{-3}).$$
305. $$ 4+2\sqrt{-3} = (1-1\sqrt{-3})(-1+1\sqrt{-3}) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
306. $$ 4+2\sqrt{-3} = (1+1\sqrt{-3})(2-1\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
307. $$ 4+4\sqrt{-3} = (-4+4\sqrt{-3})(-1\sqrt{-3}) + (-8),$$
     $$ N(-4+4\sqrt{-3}) = N(-8).$$
308. $$ 4+4\sqrt{-3} = (4-4\sqrt{-3})(-1) + (8),$$
     $$ N(4-4\sqrt{-3}) = N(8).$$
309. $$ 5-5\sqrt{-3} = (-2)(-3+2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
310. $$ 5-5\sqrt{-3} = (-1-1\sqrt{-3})(2+2\sqrt{-3}) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
311. $$ 5-5\sqrt{-3} = (1+1\sqrt{-3})(-3-3\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
312. $$ 5-5\sqrt{-3} = (2)(2-3\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
313. $$ 5-5\sqrt{-3} = (5+5\sqrt{-3})(-1-1\sqrt{-3}) + (-5+5\sqrt{-3}),$$
     $$ N(5+5\sqrt{-3}) = N(-5+5\sqrt{-3}).$$
314. $$ 5-3\sqrt{-3} = (-2)(-3+1\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
315. $$ 5-3\sqrt{-3} = (-1+1\sqrt{-3})(-4-1\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
316. $$ 5-3\sqrt{-3} = (1-1\sqrt{-3})(3) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
317. $$ 5-3\sqrt{-3} = (2)(2-2\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
318. $$ 5-3\sqrt{-3} = (2+4\sqrt{-3})(-1-1\sqrt{-3}) + (-5+3\sqrt{-3}),$$
     $$ N(2+4\sqrt{-3}) = N(-5+3\sqrt{-3}).$$
319. $$ 5-1\sqrt{-3} = (-4-2\sqrt{-3})(-1) + (1-3\sqrt{-3}),$$
     $$ N(-4-2\sqrt{-3}) = N(1-3\sqrt{-3}).$$
320. $$ 5-1\sqrt{-3} = (-2)(-3) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
321. $$ 5-1\sqrt{-3} = (-1-1\sqrt{-3})(-1+1\sqrt{-3}) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
322. $$ 5-1\sqrt{-3} = (-1+3\sqrt{-3})(-1-1\sqrt{-3}) + (-5+1\sqrt{-3}),$$
     $$ N(-1+3\sqrt{-3}) = N(-5+1\sqrt{-3}).$$
323. $$ 5-1\sqrt{-3} = (1+1\sqrt{-3})(-2\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
324. $$ 5-1\sqrt{-3} = (2)(2-1\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
325. $$ 5-1\sqrt{-3} = (4+2\sqrt{-3})(-1\sqrt{-3}) + (-1+3\sqrt{-3}),$$
     $$ N(4+2\sqrt{-3}) = N(-1+3\sqrt{-3}).$$
326. $$ 5+1\sqrt{-3} = (-4+2\sqrt{-3})(-1-1\sqrt{-3}) + (-5-1\sqrt{-3}),$$
     $$ N(-4+2\sqrt{-3}) = N(-5-1\sqrt{-3}).$$
327. $$ 5+1\sqrt{-3} = (-2)(-3-1\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
328. $$ 5+1\sqrt{-3} = (-1-3\sqrt{-3})(-1) + (4-2\sqrt{-3}),$$
     $$ N(-1-3\sqrt{-3}) = N(4-2\sqrt{-3}).$$
329. $$ 5+1\sqrt{-3} = (-1+1\sqrt{-3})(-1-2\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
330. $$ 5+1\sqrt{-3} = (1-1\sqrt{-3})(1\sqrt{-3}) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
331. $$ 5+1\sqrt{-3} = (1+3\sqrt{-3})(-1\sqrt{-3}) + (-4+2\sqrt{-3}),$$
     $$ N(1+3\sqrt{-3}) = N(-4+2\sqrt{-3}).$$
332. $$ 5+1\sqrt{-3} = (2)(2) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
333. $$ 5+3\sqrt{-3} = (-2)(-3-2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
334. $$ 5+3\sqrt{-3} = (-2+4\sqrt{-3})(-1\sqrt{-3}) + (-7+1\sqrt{-3}),$$
     $$ N(-2+4\sqrt{-3}) = N(-7+1\sqrt{-3}).$$
335. $$ 5+3\sqrt{-3} = (-1-1\sqrt{-3})(-4) + (1-1\sqrt{-3}),$$
     $$ N(-1-1\sqrt{-3}) = N(1-1\sqrt{-3}).$$
336. $$ 5+3\sqrt{-3} = (1+1\sqrt{-3})(3-1\sqrt{-3}) + (-1+1\sqrt{-3}),$$
     $$ N(1+1\sqrt{-3}) = N(-1+1\sqrt{-3}).$$
337. $$ 5+3\sqrt{-3} = (2-4\sqrt{-3})(-1) + (7-1\sqrt{-3}),$$
     $$ N(2-4\sqrt{-3}) = N(7-1\sqrt{-3}).$$
338. $$ 5+3\sqrt{-3} = (2)(2+1\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
339. $$ 5+5\sqrt{-3} = (-5+5\sqrt{-3})(-1\sqrt{-3}) + (-10),$$
     $$ N(-5+5\sqrt{-3}) = N(-10).$$
340. $$ 5+5\sqrt{-3} = (-2)(-3-3\sqrt{-3}) + (-1-1\sqrt{-3}),$$
     $$ N(-2) = N(-1-1\sqrt{-3}).$$
341. $$ 5+5\sqrt{-3} = (-1+1\sqrt{-3})(2-3\sqrt{-3}) + (-2),$$
     $$ N(-1+1\sqrt{-3}) = N(-2).$$
342. $$ 5+5\sqrt{-3} = (1-1\sqrt{-3})(-3+2\sqrt{-3}) + (2),$$
     $$ N(1-1\sqrt{-3}) = N(2).$$
343. $$ 5+5\sqrt{-3} = (2)(2+2\sqrt{-3}) + (1+1\sqrt{-3}),$$
     $$ N(2) = N(1+1\sqrt{-3}).$$
344. $$ 5+5\sqrt{-3} = (5-5\sqrt{-3})(-1) + (10),$$
     $$ N(5-5\sqrt{-3}) = N(10).$$

И ещё один скрипт, где реализован алгоритм деления с остатком в $R$:

<?php

class Element
{
    public int $a;
    public int $b;

    function __construct(int $a, int $b)
    {
        $this->a = $a;
        $this->b = $b;
    }

    function multiply(Element $e): Element
    {
        return new Element($this->a * $e->a - 3 * $this->b * $e->b, $this->a * $e->b + $this->b * $e->a);
    }

    function add(Element $e): Element
    {
        return new Element($this->a + $e->a, $this->b + $e->b);
    }

    function subtract(Element $e): Element
    {
        return new Element($this->a - $e->a, $this->b - $e->b);
    }

    function getNorm(): int
    {
        return $this->a ** 2 + 3 * $this->b ** 2;
    }

    function __toString(): string
    {
        $sign = $this->b <= 0 ? '' : '+';
        $imaginary = $this->b === 0 ? '': "{$this->b}\sqrt{-3}";

        if ($this->a === 0 && $this->b !== 0) {
            return $imaginary;
        } else {
            return "{$this->a}{$sign}{$imaginary}";
        }
    }

    function equals(Element $e)
    {
        return $this->a === $e->a && $this->b === $e->b;
    }

    function divide(Element $d)
    {
        $denom = $d->a ** 2 + 3 * $d->b ** 2;
        $numA = $this->a * $d->a + 3 * $this->b * $d->b;
        
        $numB = $this->b * $d->a - $this->a * $d->b;

        $getClosestInt = function ($num, $denom) {

            $da = intdiv($num, $denom);
            $ra = $num % $denom;

            if (abs($ra)*2 < $denom) {
                return $da;
            } elseif (abs($ra)*2 === $denom) {
                return $da;
            } else {
                return $num < 0 ? $da - 1 : ($num > 0 ? $da + 1 : $num);
            }
        };

        return new Element($getClosestInt($numA, $denom), $getClosestInt($numB, $denom));
    }
}

$N = 5;

for ($a = -$N; $a <= $N; $a++) {
    for ($b = -$N; $b <= $N; $b++) {
        if ($a === 0 && $b === 0) {
            continue;
        }
        for ($c = -$N; $c <= $N; $c++) {
            for ($d = -$N; $d <= $N; $d++) {
                if ($c === 0 && $d === 0) {
                    continue;
                }

                $alpha  = new Element($a, $b);
                $beta   = new Element($c, $d);
                $lambda = $alpha->divide($beta);

                $r = $alpha->subtract($beta->multiply($lambda));

                $betaNorm = $beta->getNorm();
                $rNorm = $r->getNorm();
                $sign = $betaNorm < $rNorm ? '<' : ($betaNorm === $rNorm ? '=' : '>');

                echo "<li>\n";
                echo "$$ $alpha = ($beta)($lambda) + ($r),$$\n";
                echo "$$ $betaNorm $sign $rNorm.$$\n";
                echo "$$ N($beta) $sign N($r).$$\n";
                echo "</li>\n";
            }
        }
    }
}

Часть его вывода:

1. $$ -5-5\sqrt{-3} = (-5-5\sqrt{-3})(1) + (0),$$
   $$ 100 > 0.$$
   $$ N(-5-5\sqrt{-3}) > N(0).$$
2. $$ -5-5\sqrt{-3} = (-5-4\sqrt{-3})(1) + (-1\sqrt{-3}),$$
   $$ 73 > 3.$$
   $$ N(-5-4\sqrt{-3}) > N(-1\sqrt{-3}).$$
3. $$ -5-5\sqrt{-3} = (-5-3\sqrt{-3})(1) + (-2\sqrt{-3}),$$
   $$ 52 > 12.$$
   $$ N(-5-3\sqrt{-3}) > N(-2\sqrt{-3}).$$
4. $$ -5-5\sqrt{-3} = (-5-2\sqrt{-3})(1) + (-3\sqrt{-3}),$$
   $$ 37 > 27.$$
   $$ N(-5-2\sqrt{-3}) > N(-3\sqrt{-3}).$$
5. $$ -5-5\sqrt{-3} = (-5-1\sqrt{-3})(1+1\sqrt{-3}) + (-3+1\sqrt{-3}),$$
   $$ 28 > 12.$$
   $$ N(-5-1\sqrt{-3}) > N(-3+1\sqrt{-3}).$$
6. $$ -5-5\sqrt{-3} = (-5)(1+1\sqrt{-3}) + (0),$$
   $$ 25 > 0.$$
   $$ N(-5) > N(0).$$
7. $$ -5-5\sqrt{-3} = (-5+1\sqrt{-3})(1\sqrt{-3}) + (-2),$$
   $$ 28 > 4.$$
   $$ N(-5+1\sqrt{-3}) > N(-2).$$
8. $$ -5-5\sqrt{-3} = (-5+2\sqrt{-3})(1\sqrt{-3}) + (1),$$
   $$ 37 > 1.$$
   $$ N(-5+2\sqrt{-3}) > N(1).$$
9. $$ -5-5\sqrt{-3} = (-5+3\sqrt{-3})(1\sqrt{-3}) + (4),$$
   $$ 52 > 16.$$
   $$ N(-5+3\sqrt{-3}) > N(4).$$
10. $$ -5-5\sqrt{-3} = (-5+4\sqrt{-3})(1\sqrt{-3}) + (7),$$
    $$ 73 > 49.$$
    $$ N(-5+4\sqrt{-3}) > N(7).$$
11. $$ -5-5\sqrt{-3} = (-5+5\sqrt{-3})(0) + (-5-5\sqrt{-3}),$$
    $$ 100 = 100.$$
    $$ N(-5+5\sqrt{-3}) = N(-5-5\sqrt{-3}).$$
12. $$ -5-5\sqrt{-3} = (-4-5\sqrt{-3})(1) + (-1),$$
    $$ 91 > 1.$$
    $$ N(-4-5\sqrt{-3}) > N(-1).$$
13. $$ -5-5\sqrt{-3} = (-4-4\sqrt{-3})(1) + (-1-1\sqrt{-3}),$$
    $$ 64 > 4.$$
    $$ N(-4-4\sqrt{-3}) > N(-1-1\sqrt{-3}).$$
14. $$ -5-5\sqrt{-3} = (-4-3\sqrt{-3})(2) + (3+1\sqrt{-3}),$$
    $$ 43 > 12.$$
    $$ N(-4-3\sqrt{-3}) > N(3+1\sqrt{-3}).$$
15. $$ -5-5\sqrt{-3} = (-4-2\sqrt{-3})(2) + (3-1\sqrt{-3}),$$
    $$ 28 > 12.$$
    $$ N(-4-2\sqrt{-3}) > N(3-1\sqrt{-3}).$$
16. $$ -5-5\sqrt{-3} = (-4-1\sqrt{-3})(2+1\sqrt{-3}) + (1\sqrt{-3}),$$
    $$ 19 > 3.$$
    $$ N(-4-1\sqrt{-3}) > N(1\sqrt{-3}).$$
17. $$ -5-5\sqrt{-3} = (-4)(1+1\sqrt{-3}) + (-1-1\sqrt{-3}),$$
    $$ 16 > 4.$$
    $$ N(-4) > N(-1-1\sqrt{-3}).$$
18. $$ -5-5\sqrt{-3} = (-4+1\sqrt{-3})(1\sqrt{-3}) + (-2-1\sqrt{-3}),$$
    $$ 19 > 7.$$
    $$ N(-4+1\sqrt{-3}) > N(-2-1\sqrt{-3}).$$
19. $$ -5-5\sqrt{-3} = (-4+2\sqrt{-3})(1\sqrt{-3}) + (1-1\sqrt{-3}),$$
    $$ 28 > 4.$$
    $$ N(-4+2\sqrt{-3}) > N(1-1\sqrt{-3}).$$
20. $$ -5-5\sqrt{-3} = (-4+3\sqrt{-3})(-1+1\sqrt{-3}) + (2\sqrt{-3}),$$
    $$ 43 > 12.$$
    $$ N(-4+3\sqrt{-3}) > N(2\sqrt{-3}).$$
21. $$ -5-5\sqrt{-3} = (-4+4\sqrt{-3})(-1+1\sqrt{-3}) + (3+3\sqrt{-3}),$$
    $$ 64 > 36.$$
    $$ N(-4+4\sqrt{-3}) > N(3+3\sqrt{-3}).$$
22. $$ -5-5\sqrt{-3} = (-4+5\sqrt{-3})(-1) + (-9),$$
    $$ 91 > 81.$$
    $$ N(-4+5\sqrt{-3}) > N(-9).$$
23. $$ -5-5\sqrt{-3} = (-3-5\sqrt{-3})(1) + (-2),$$
    $$ 84 > 4.$$
    $$ N(-3-5\sqrt{-3}) > N(-2).$$
24. $$ -5-5\sqrt{-3} = (-3-4\sqrt{-3})(1) + (-2-1\sqrt{-3}),$$
    $$ 57 > 7.$$
    $$ N(-3-4\sqrt{-3}) > N(-2-1\sqrt{-3}).$$
25. $$ -5-5\sqrt{-3} = (-3-3\sqrt{-3})(2) + (1+1\sqrt{-3}),$$
    $$ 36 > 4.$$
    $$ N(-3-3\sqrt{-3}) > N(1+1\sqrt{-3}).$$
26. $$ -5-5\sqrt{-3} = (-3-2\sqrt{-3})(2) + (1-1\sqrt{-3}),$$
    $$ 21 > 4.$$
    $$ N(-3-2\sqrt{-3}) > N(1-1\sqrt{-3}).$$
27. $$ -5-5\sqrt{-3} = (-3-1\sqrt{-3})(2+1\sqrt{-3}) + (-2),$$
    $$ 12 > 4.$$
    $$ N(-3-1\sqrt{-3}) > N(-2).$$
28. $$ -5-5\sqrt{-3} = (-3)(2+2\sqrt{-3}) + (1+1\sqrt{-3}),$$
    $$ 9 > 4.$$
    $$ N(-3) > N(1+1\sqrt{-3}).$$
29. $$ -5-5\sqrt{-3} = (-3+1\sqrt{-3})(2\sqrt{-3}) + (1+1\sqrt{-3}),$$
    $$ 12 > 4.$$
    $$ N(-3+1\sqrt{-3}) > N(1+1\sqrt{-3}).$$
30. $$ -5-5\sqrt{-3} = (-3+2\sqrt{-3})(-1+1\sqrt{-3}) + (-2),$$
    $$ 21 > 4.$$
    $$ N(-3+2\sqrt{-3}) > N(-2).$$
31. $$ -5-5\sqrt{-3} = (-3+3\sqrt{-3})(-1+1\sqrt{-3}) + (1+1\sqrt{-3}),$$
    $$ 36 > 4.$$
    $$ N(-3+3\sqrt{-3}) > N(1+1\sqrt{-3}).$$
32. $$ -5-5\sqrt{-3} = (-3+4\sqrt{-3})(-1+1\sqrt{-3}) + (4+2\sqrt{-3}),$$
    $$ 57 > 28.$$
    $$ N(-3+4\sqrt{-3}) > N(4+2\sqrt{-3}).$$
33. $$ -5-5\sqrt{-3} = (-3+5\sqrt{-3})(-1) + (-8),$$
    $$ 84 > 64.$$
    $$ N(-3+5\sqrt{-3}) > N(-8).$$
34. $$ -5-5\sqrt{-3} = (-2-5\sqrt{-3})(1) + (-3),$$
    $$ 79 > 9.$$
    $$ N(-2-5\sqrt{-3}) > N(-3).$$
35. $$ -5-5\sqrt{-3} = (-2-4\sqrt{-3})(1) + (-3-1\sqrt{-3}),$$
    $$ 52 > 12.$$
    $$ N(-2-4\sqrt{-3}) > N(-3-1\sqrt{-3}).$$
36. $$ -5-5\sqrt{-3} = (-2-3\sqrt{-3})(2) + (-1+1\sqrt{-3}),$$
    $$ 31 > 4.$$
    $$ N(-2-3\sqrt{-3}) > N(-1+1\sqrt{-3}).$$
37. $$ -5-5\sqrt{-3} = (-2-2\sqrt{-3})(2) + (-1-1\sqrt{-3}),$$
    $$ 16 > 4.$$
    $$ N(-2-2\sqrt{-3}) > N(-1-1\sqrt{-3}).$$
38. $$ -5-5\sqrt{-3} = (-2-1\sqrt{-3})(4+1\sqrt{-3}) + (1\sqrt{-3}),$$
    $$ 7 > 3.$$
    $$ N(-2-1\sqrt{-3}) > N(1\sqrt{-3}).$$
39. $$ -5-5\sqrt{-3} = (-2)(2+2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
    $$ 4 = 4.$$
    $$ N(-2) = N(-1-1\sqrt{-3}).$$
40. $$ -5-5\sqrt{-3} = (-2+1\sqrt{-3})(-1+2\sqrt{-3}) + (-1),$$
    $$ 7 > 1.$$
    $$ N(-2+1\sqrt{-3}) > N(-1).$$
41. $$ -5-5\sqrt{-3} = (-2+2\sqrt{-3})(-1+1\sqrt{-3}) + (-1-1\sqrt{-3}),$$
    $$ 16 > 4.$$
    $$ N(-2+2\sqrt{-3}) > N(-1-1\sqrt{-3}).$$
42. $$ -5-5\sqrt{-3} = (-2+3\sqrt{-3})(-1+1\sqrt{-3}) + (2),$$
    $$ 31 > 4.$$
    $$ N(-2+3\sqrt{-3}) > N(2).$$
43. $$ -5-5\sqrt{-3} = (-2+4\sqrt{-3})(-1+1\sqrt{-3}) + (5+1\sqrt{-3}),$$
    $$ 52 > 28.$$
    $$ N(-2+4\sqrt{-3}) > N(5+1\sqrt{-3}).$$
44. $$ -5-5\sqrt{-3} = (-2+5\sqrt{-3})(-1) + (-7),$$
    $$ 79 > 49.$$
    $$ N(-2+5\sqrt{-3}) > N(-7).$$
45. $$ -5-5\sqrt{-3} = (-1-5\sqrt{-3})(1) + (-4),$$
    $$ 76 > 16.$$
    $$ N(-1-5\sqrt{-3}) > N(-4).$$
46. $$ -5-5\sqrt{-3} = (-1-4\sqrt{-3})(1) + (-4-1\sqrt{-3}),$$
    $$ 49 > 19.$$
    $$ N(-1-4\sqrt{-3}) > N(-4-1\sqrt{-3}).$$
47. $$ -5-5\sqrt{-3} = (-1-3\sqrt{-3})(2) + (-3+1\sqrt{-3}),$$
    $$ 28 > 12.$$
    $$ N(-1-3\sqrt{-3}) > N(-3+1\sqrt{-3}).$$
48. $$ -5-5\sqrt{-3} = (-1-2\sqrt{-3})(3) + (-2+1\sqrt{-3}),$$
    $$ 13 > 7.$$
    $$ N(-1-2\sqrt{-3}) > N(-2+1\sqrt{-3}).$$
49. $$ -5-5\sqrt{-3} = (-1-1\sqrt{-3})(5) + (0),$$
    $$ 4 > 0.$$
    $$ N(-1-1\sqrt{-3}) > N(0).$$
50. $$ -5-5\sqrt{-3} = (-1)(5+5\sqrt{-3}) + (0),$$
    $$ 1 > 0.$$
    $$ N(-1) > N(0).$$
51. $$ -5-5\sqrt{-3} = (-1+1\sqrt{-3})(-2+2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
    $$ 4 = 4.$$
    $$ N(-1+1\sqrt{-3}) = N(-1-1\sqrt{-3}).$$
52. $$ -5-5\sqrt{-3} = (-1+2\sqrt{-3})(-2+1\sqrt{-3}) + (-1),$$
    $$ 13 > 1.$$
    $$ N(-1+2\sqrt{-3}) > N(-1).$$
53. $$ -5-5\sqrt{-3} = (-1+3\sqrt{-3})(-1+1\sqrt{-3}) + (3-1\sqrt{-3}),$$
    $$ 28 > 12.$$
    $$ N(-1+3\sqrt{-3}) > N(3-1\sqrt{-3}).$$
54. $$ -5-5\sqrt{-3} = (-1+4\sqrt{-3})(-1+1\sqrt{-3}) + (6),$$
    $$ 49 > 36.$$
    $$ N(-1+4\sqrt{-3}) > N(6).$$
55. $$ -5-5\sqrt{-3} = (-1+5\sqrt{-3})(-1) + (-6),$$
    $$ 76 > 36.$$
    $$ N(-1+5\sqrt{-3}) > N(-6).$$
56. $$ -5-5\sqrt{-3} = (-5\sqrt{-3})(1) + (-5),$$
    $$ 75 > 25.$$
    $$ N(-5\sqrt{-3}) > N(-5).$$
57. $$ -5-5\sqrt{-3} = (-4\sqrt{-3})(1) + (-5-1\sqrt{-3}),$$
    $$ 48 > 28.$$
    $$ N(-4\sqrt{-3}) > N(-5-1\sqrt{-3}).$$
58. $$ -5-5\sqrt{-3} = (-3\sqrt{-3})(2-1\sqrt{-3}) + (4+1\sqrt{-3}),$$
    $$ 27 > 19.$$
    $$ N(-3\sqrt{-3}) > N(4+1\sqrt{-3}).$$
59. $$ -5-5\sqrt{-3} = (-2\sqrt{-3})(2-1\sqrt{-3}) + (1-1\sqrt{-3}),$$
    $$ 12 > 4.$$
    $$ N(-2\sqrt{-3}) > N(1-1\sqrt{-3}).$$
60. $$ -5-5\sqrt{-3} = (-1\sqrt{-3})(5-2\sqrt{-3}) + (1),$$
    $$ 3 > 1.$$
    $$ N(-1\sqrt{-3}) > N(1).$$
61. $$ -5-5\sqrt{-3} = (1\sqrt{-3})(-5+2\sqrt{-3}) + (1),$$
    $$ 3 > 1.$$
    $$ N(1\sqrt{-3}) > N(1).$$
62. $$ -5-5\sqrt{-3} = (2\sqrt{-3})(-2+1\sqrt{-3}) + (1-1\sqrt{-3}),$$
    $$ 12 > 4.$$
    $$ N(2\sqrt{-3}) > N(1-1\sqrt{-3}).$$
63. $$ -5-5\sqrt{-3} = (3\sqrt{-3})(-2+1\sqrt{-3}) + (4+1\sqrt{-3}),$$
    $$ 27 > 19.$$
    $$ N(3\sqrt{-3}) > N(4+1\sqrt{-3}).$$
64. $$ -5-5\sqrt{-3} = (4\sqrt{-3})(-1) + (-5-1\sqrt{-3}),$$
    $$ 48 > 28.$$
    $$ N(4\sqrt{-3}) > N(-5-1\sqrt{-3}).$$
65. $$ -5-5\sqrt{-3} = (5\sqrt{-3})(-1) + (-5),$$
    $$ 75 > 25.$$
    $$ N(5\sqrt{-3}) > N(-5).$$
66. $$ -5-5\sqrt{-3} = (1-5\sqrt{-3})(1) + (-6),$$
    $$ 76 > 36.$$
    $$ N(1-5\sqrt{-3}) > N(-6).$$
67. $$ -5-5\sqrt{-3} = (1-4\sqrt{-3})(1-1\sqrt{-3}) + (6),$$
    $$ 49 > 36.$$
    $$ N(1-4\sqrt{-3}) > N(6).$$
68. $$ -5-5\sqrt{-3} = (1-3\sqrt{-3})(1-1\sqrt{-3}) + (3-1\sqrt{-3}),$$
    $$ 28 > 12.$$
    $$ N(1-3\sqrt{-3}) > N(3-1\sqrt{-3}).$$
69. $$ -5-5\sqrt{-3} = (1-2\sqrt{-3})(2-1\sqrt{-3}) + (-1),$$
    $$ 13 > 1.$$
    $$ N(1-2\sqrt{-3}) > N(-1).$$
70. $$ -5-5\sqrt{-3} = (1-1\sqrt{-3})(2-2\sqrt{-3}) + (-1-1\sqrt{-3}),$$
    $$ 4 = 4.$$
    $$ N(1-1\sqrt{-3}) = N(-1-1\sqrt{-3}).$$
71. $$ -5-5\sqrt{-3} = (1)(-5-5\sqrt{-3}) + (0),$$
    $$ 1 > 0.$$
    $$ N(1) > N(0).$$
72. $$ -5-5\sqrt{-3} = (1+1\sqrt{-3})(-5) + (0),$$
    $$ 4 > 0.$$
    $$ N(1+1\sqrt{-3}) > N(0).$$
73. $$ -5-5\sqrt{-3} = (1+2\sqrt{-3})(-3) + (-2+1\sqrt{-3}),$$
    $$ 13 > 7.$$
    $$ N(1+2\sqrt{-3}) > N(-2+1\sqrt{-3}).$$
74. $$ -5-5\sqrt{-3} = (1+3\sqrt{-3})(-2) + (-3+1\sqrt{-3}),$$
    $$ 28 > 12.$$
    $$ N(1+3\sqrt{-3}) > N(-3+1\sqrt{-3}).$$
75. $$ -5-5\sqrt{-3} = (1+4\sqrt{-3})(-1) + (-4-1\sqrt{-3}),$$
    $$ 49 > 19.$$
    $$ N(1+4\sqrt{-3}) > N(-4-1\sqrt{-3}).$$