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. $$ -5-5\sqrt{-3} = (1+4\sqrt{-3})(-1) + (-4-1\sqrt{-3}),$$
    $$ 49 > 19.$$
    $$ N(1+4\sqrt{-3}) > N(-4-1\sqrt{-3}).$$