If output_opt is set to output = product, then the return value is of the form $\pm u^a{p}_1^{b_1}\cdots p_n^{b_n}$ where the $p_i$ are distinct prime factors and the $b_i$ are positive integers.

If $d\>0$, then $u$ is either $w$ or $\stackrel{\&conjugate0;}{w}$ where $w$ is the fundamental unit in $Z\left(\sqrt{d}\right)$ and $a$ is a non-negative integer.

If $d<0$, then $u$ is a unit in $Z\left(\sqrt{d}\right)$ and $a=1$.

If output_opt is set to output = list, then the return value is of the form $\left[\left[s\&comma;x\&comma;y\&comma;a\right]\&comma;\left[f_1\&comma;\dots \&comma;f_n\right]\right]$ where $s\&equals;\pm 1$ and each $f_i$ is a three element list of the form $\left[p\&comma;q\&comma;k\right]$. Each $p\&plus;q\sqrt{d}$ is a distinct prime and $k$ is a positive integer.

If $d\&gt;0$, then $u\&equals;x\&plus;y\sqrt{d}$ where $u$ is as previously described and $a$ is a non-negative integer.

If $d<0$, then $x\&plus;y\sqrt{d}$ is a unit in $Z\left(\sqrt{d}\right)$. Let $t\&equals;x\&plus;y\sqrt{d}$. If $t\&equals;\pm 1$ then $t\&equals;s$ and $x\&comma;y\&comma;a\&equals;1\&comma;0\&comma;0$. Otherwise, $s\&comma;a\&equals;1\&comma;1$.

The FactorNormEuclidean function computes the integer factorization of z in the ring of integers $Z\left(\sqrt{d}\right)$ of the quadratic field $Q\left(\sqrt{d}\right)$.

Consider the absolute value of the field norm of $Q\left(\sqrt{d}\right)$ as a field extension of $Q$, denoted by $N$. If d is one of $\mathrm{-11}\&comma;\mathrm{-7}\&comma;\mathrm{-3}\&comma;\mathrm{-2}\&comma;\mathrm{-1}\&comma;2\&comma;3\&comma;5\&comma;6\&comma;7\&comma;11\&comma;13\&comma;17\&comma;19\&comma;21\&comma;29\&comma;33\&comma;37\&comma;41\&comma;57\&comma;73$, then $N$ satisfies the following property. If $a$ and $b$ are in $Q\left(\sqrt{d}\right)$ and $b\ne 0$, then there exists $q$ and $r$ in $Q\left(\sqrt{d}\right)$ such that $a\&equals;bq\&plus;r$ and $N\left(r\right)<N\left(b\right)$. In this case, $N$ is said to be a Euclidean function on $Q\left(\sqrt{d}\right)$ and $Q\left(\sqrt{d}\right)$ is said to be a norm-Euclidean field.

When $d=2,3\mathbf{mod}4$, integers in $Z\left(\sqrt{d}\right)$ have the form $a\&plus;b\sqrt{d}$ and when $d=1\mathbf{mod}4$ they have the form $a\&plus;b\left(\frac{1}{2}\&plus;\frac{1}{2}\sqrt{d}\right)$, where $a$ and $b$ are rational integers. Alternatively for when $d=1\mathbf{mod}4$, integers have the form $\frac{a}{2}\&plus;\frac{b}{2}\sqrt{d}$ where $a$ and $b$ are rational integers of the same parity.

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

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

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

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

$38477343$

$\mathrm{FactorNormEuclidean}\left(38434\sqrt{33}\&comma;33\&comma;\mathrm{output}\&equals;\mathrm{product}\right)$

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

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

$38434\sqrt{33}$

$\mathrm{FactorNormEuclidean}\left(408294234124-4242\sqrt{29}\&comma;29\&comma;\mathrm{output}\&equals;\mathrm{list}\right)$

$\left[\left[\mathrm{-1}\&comma;\frac{5}{2}\&comma;-\frac{1}{2}\&comma;4\right]\&comma;\left[\left[2\&comma;0\&comma;1\right]\&comma;\left[4\&comma;\mathrm{-1}\&comma;0\right]\&comma;\left[4\&comma;1\&comma;1\right]\&comma;\left[11\&comma;\mathrm{-2}\&comma;0\right]\&comma;\left[11\&comma;2\&comma;1\right]\&comma;\left[38\&comma;\mathrm{-7}\&comma;0\right]\&comma;\left[38\&comma;7\&comma;1\right]\&comma;\left[\frac{1}{2}\&comma;-\frac{1}{2}\&comma;1\right]\&comma;\left[\frac{1}{2}\&comma;\frac{1}{2}\&comma;1\right]\&comma;\left[\frac{955872689}{2}\&comma;-\frac{331629325}{2}\&comma;0\right]\&comma;\left[\frac{955872689}{2}\&comma;\frac{331629325}{2}\&comma;1\right]\right]\right]$

$\mathrm{FactorNormEuclidean}\left(\frac{3}{2}\&comma;2\right)$

Error, (in NumberTheory:-FactorNormEuclidean) 3/2 is not an integer in Q(sqrt(2))

The NumberTheory[FactorNormEuclidean] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.