Important: The numtheory package has been deprecated.  Use the superseding command NumberTheory[FactorNormEuclidean] instead.

The factorEQ function returns the integer factorization of m in the Euclidean ring $Z\left(\sqrt{d}\right)$.

Given integers $a$ and $b$ of $Z\left(\sqrt{d}\right)$, with $b\ne 0$, there is an integer $q$ such that $a=qb+r$, $\mathrm{norm}\left(r\right)<\mathrm{norm}\left(b\right)$ is true in $Z\left(\sqrt{d}\right)$. In these circumstances we say that there is a Euclidean algorithm in $Z\left(\sqrt{d}\right)$ and that the ring is Euclidean.

Euclidean quadratic number fields have been completely determined. They are $Z\left(\sqrt{d}\right)$ where d = -1, -2, -3, -7, -11, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, and 73.

When $d=2$,$3\mathbf{mod}4$, all integers of $Z\left(\sqrt{d}\right)$ have the form $a+b\sqrt{d}$, where $a$ and $b$ are rational integers. When $d=1\mathbf{mod}4$, all integers of $Z\left(\sqrt{d}\right)$ are of the form $\frac{\left(a+b\sqrt{d}\right)}{2}$ where $a$ and $b$ are rational integers and of the same parity.

The answer is in the form: $\pm 1u{\left(f_1\right)}^{e_1}\mathrm{...}{\left(f_{\mathrm{n}}\right)}^{e_{\mathrm{n}}}$ such that $m=\pm 1\cdot u{f}_1^{e_1}\dots f_n^{e_n}$ where $f_1,\dots ,f_n$ are distinct prime factors of m, $e_1,\dots ,e_n$ are non-negative integer numbers, $u$ is a unit in $Z\left(\sqrt{d}\right)$. For real Euclidean quadratic rings, i.e.  d > 0, $u$ is represented under the form $w^n$ or ${{\stackrel{\&conjugate0;}{w}}^{}}^n$ or ${\left(-w\right)}^n$ or ${\left(-\stackrel{\&conjugate0;}{w}\right)}^n$ where $w$ is the fundamental unit, and $n$ is a positive integer.

The expand function may be applied to cause the factors to be multiplied together again.

$\mathrm{with}\left(\mathrm{numtheory}\right)\&colon;$

$\mathrm{factorEQ}\left(38477343\&comma;11\right)$

$\left(3\right)\left(125+34\sqrt{11}\right)\left(125-34\sqrt{11}\right)\left(85+16\sqrt{11}\right)\left(85-16\sqrt{11}\right)$

$\mathrm{expand}\left(\right)$

$38477343$

$\mathrm{factorEQ}\left(38434\mathrm{sqrt}\left(33\right)\&comma;33\right)$

$\left(\sqrt{33}\right)\left(-23+4\sqrt{33}\right)\left(\frac{5}{2}+\frac{\sqrt{33}}{2}\right)\left(\frac{5}{2}-\frac{\sqrt{33}}{2}\right){\left(11+2\sqrt{33}\right)}^2\left(58+7\sqrt{33}\right)\left(58-7\sqrt{33}\right)$

$\mathrm{expand}\left(\right)$

$38434\sqrt{33}$

$\mathrm{factorEQ}\left(408294234124-4242\mathrm{sqrt}\left(29\right)\&comma;29\right)$

$-\left(2\right)\left(\frac{1}{2}+\frac{\sqrt{29}}{2}\right)\left(\frac{1}{2}-\frac{\sqrt{29}}{2}\right){\left(\frac{5}{2}-\frac{\sqrt{29}}{2}\right)}^4\left(11+2\sqrt{29}\right)\left(4+\sqrt{29}\right)\left(38+7\sqrt{29}\right)\left(\frac{955872689}{2}+\frac{331629325\sqrt{29}}{2}\right)$

$\mathrm{expand}\left(\right)$

$408294234124-4242\sqrt{29}$