numtheory/factorEQ(deprecated) - Help

numtheory

 factorEQ
 integer factorization in Z(sqrt(d)) where Z(sqrt(d)) is a Euclidean ring

 Calling Sequence factorEQ(m, d)

Parameters

 m - integer, list or set of integers in $Z\left(\sqrt{d}\right)$ d - integer where $\sqrt{d}$ is a Euclidean ring

Description

 • Important: The numtheory[factorEQ] command 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 $a=bq+r$, $\mathrm{norm}\left(r\right)<\mathrm{norm}\left(b\right)$ 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 $d=2$,$3\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}4$, all integers of $Z\left(\sqrt{d}\right)$ have the form $a+b\sqrt{d}$, where $a$ and $b$ are rational integers. When $d=1\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}4$, all integers of $Z\left(\sqrt{d}\right)$ are of the form $\frac{1}{2}\left(a+b\sqrt{d}\right)$ where $a$ and $b$ are rational integers and of the same parity.
 • The answer is in the form: $±1u{\left({f}_{1}\right)}^{{e}_{1}}\mathrm{...}{\left({f}_{n}\right)}^{{e}_{n}}$ such that $m=±1\cdot u{f}_{1}^{{e}_{1}}\dots {f}_{n}^{{e}_{n}}$ 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.

Examples

 > $\mathrm{with}\left(\mathrm{numtheory}\right):$
 > $\mathrm{factorEQ}\left(38477343,11\right)$
 ${}\left({3}\right){}{}\left({125}{+}{34}{}\sqrt{{11}}\right){}{}\left({125}{-}{34}{}\sqrt{{11}}\right){}{}\left({85}{+}{16}{}\sqrt{{11}}\right){}{}\left({85}{-}{16}{}\sqrt{{11}}\right)$ (1)
 > $\mathrm{expand}\left(\right)$
 ${38477343}$ (2)
 > $\mathrm{factorEQ}\left(38434\sqrt{33},33\right)$
 ${}\left(\sqrt{{33}}\right){}{}\left({-}{23}{+}{4}{}\sqrt{{33}}\right){}{}\left(\frac{{5}}{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{33}}\right){}{}\left(\frac{{5}}{{2}}{-}\frac{{1}}{{2}}{}\sqrt{{33}}\right){}{{}\left({11}{+}{2}{}\sqrt{{33}}\right)}^{{2}}{}{}\left({58}{+}{7}{}\sqrt{{33}}\right){}{}\left({58}{-}{7}{}\sqrt{{33}}\right)$ (3)
 > $\mathrm{expand}\left(\right)$
 ${38434}{}\sqrt{{33}}$ (4)
 > $\mathrm{factorEQ}\left(408294234124-4242\sqrt{29},29\right)$
 ${-}{}\left({2}\right){}{}\left(\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{29}}\right){}{}\left(\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}\sqrt{{29}}\right){}{{}\left(\frac{{5}}{{2}}{-}\frac{{1}}{{2}}{}\sqrt{{29}}\right)}^{{4}}{}{}\left({11}{+}{2}{}\sqrt{{29}}\right){}{}\left({4}{+}\sqrt{{29}}\right){}{}\left({38}{+}{7}{}\sqrt{{29}}\right){}{}\left(\frac{{955872689}}{{2}}{+}\frac{{331629325}}{{2}}{}\sqrt{{29}}\right)$ (5)
 > $\mathrm{expand}\left(\right)$
 ${408294234124}{-}{4242}{}\sqrt{{29}}$ (6)