 Factor - Maple Help

Ordinals

 Factor
 factor an ordinal number Calling Sequence Factor(a, output=o, form=f) Parameters

 a - ordinal, nonnegative integer, or polynomial with positive integer coefficients o - (optional) literal keyword; either list (default) or inert f - (optional) literal keyword; one of full (default), monic, rmonic or pairs Returns

 • If output=list (the default), a list of ordinals, nonnegative integers and polynomials with positive integer coefficients is returned.
 • Otherwise, if output=inert is specified, an inert product of ordinal numbers using the inert multiplication and exponentiation operators &. and &^, respectively, is returned. Factors equal to $1$ are omitted from this product representation. Description

 • The Factor(a) calling sequence computes a factored normal form of $a$ as a product of nonnegative integers and ordinals of the form ${\mathbf{\omega }}^{d}$ or ${\mathbf{\omega }}^{d}+1$.
 • If $a={\mathbf{\omega }}^{{e}_{1}}\cdot {c}_{1}+\cdots +{\mathbf{\omega }}^{{e}_{k-1}}\cdot {c}_{k-1}+{\mathbf{\omega }}^{{e}_{k}}\cdot {c}_{k}$, then the full factored normal form is:

${\mathbf{\omega }}^{{d}_{k}}\cdot {c}_{k}\cdot \left({\mathbf{\omega }}^{{d}_{k-1}}+1\right)\cdot {c}_{k-1}\cdot \dots \cdot \left({\mathbf{\omega }}^{{d}_{1}}+1\right)\cdot {c}_{1}$

 where ${d}_{k}={e}_{k}$ and ${e}_{i+1}={e}_{i}+{d}_{i}$ for $1\le i.
 • Each factor ${b}_{i}={\mathbf{\omega }}^{{d}_{i}}+1$ is irreducible in the sense that if ${b}_{i}=u\cdot v$ for some ordinals $u$ and $v$, then necessarily $u=1$ or $v=1$, and if ${b}_{i}={u}^{v}$ for some ordinals $u$ and $v$, then necessarily $u={b}_{i}$ and $v=1$.
 • The monic factored normal form is:

${\mathbf{\omega }}^{{d}_{k}}\cdot \left({\mathbf{\omega }}^{{d}_{k-1}}+{c}_{k}\right)\cdot \dots \cdot \left({\mathbf{\omega }}^{{d}_{1}}+{c}_{2}\right)\cdot {c}_{1}$

 • The rmonic factored normal form is:

${\mathbf{\omega }}^{{d}_{k}}\cdot {c}_{k}\cdot \left({\mathbf{\omega }}^{{d}_{k-1}}\cdot {c}_{k-1}+1\right)\cdot \dots \cdot \left({\mathbf{\omega }}^{{d}_{1}}\cdot {c}_{1}+1\right)$

 • If form=pairs is specified, then the result is returned in the form $\left[\left[{d}_{k},{c}_{k}\right],\left[{d}_{k-1},{c}_{k-1}\right],\mathrm{...},\left[{d}_{1},{c}_{1}\right]\right]$.
 • The ordinal $a$ can be parametric. However, unless all coefficients ${c}_{i}$ are positive when substituting arbitrary nonnegative integers for all the parameters, an error will be raised. Examples

 > $\mathrm{with}\left(\mathrm{Ordinals}\right)$
 $\left[{\mathrm{+}}{,}{\mathrm{.}}{,}{\mathrm{<}}{,}{\mathrm{<=}}{,}{\mathrm{Add}}{,}{\mathrm{Base}}{,}{\mathrm{Dec}}{,}{\mathrm{Decompose}}{,}{\mathrm{Div}}{,}{\mathrm{Eval}}{,}{\mathrm{Factor}}{,}{\mathrm{Gcd}}{,}{\mathrm{Lcm}}{,}{\mathrm{LessThan}}{,}{\mathrm{Log}}{,}{\mathrm{Max}}{,}{\mathrm{Min}}{,}{\mathrm{Mult}}{,}{\mathrm{Ordinal}}{,}{\mathrm{Power}}{,}{\mathrm{Split}}{,}{\mathrm{Sub}}{,}{\mathrm{^}}{,}{\mathrm{degree}}{,}{\mathrm{lcoeff}}{,}{\mathrm{log}}{,}{\mathrm{lterm}}{,}{\mathrm{\omega }}{,}{\mathrm{quo}}{,}{\mathrm{rem}}{,}{\mathrm{tcoeff}}{,}{\mathrm{tdegree}}{,}{\mathrm{tterm}}\right]$ (1)
 > $a≔\mathrm{Ordinal}\left(\left[\left[\mathrm{\omega },5\right],\left[9,4\right],\left[7,3\right],\left[5,3\right],\left[3,3\right],\left[2,2\right]\right]\right)$
 ${a}{≔}{{\mathbf{\omega }}}^{{\mathbf{\omega }}}{\cdot }{5}{+}{{\mathbf{\omega }}}^{{9}}{\cdot }{4}{+}{{\mathbf{\omega }}}^{{7}}{\cdot }{3}{+}{{\mathbf{\omega }}}^{{5}}{\cdot }{3}{+}{{\mathbf{\omega }}}^{{3}}{\cdot }{3}{+}{{\mathbf{\omega }}}^{{2}}{\cdot }{2}$ (2)
 > $\mathrm{Factor}\left(a\right)$
 $\left[{{\mathbf{\omega }}}^{{2}}{,}{2}{,}{\mathbf{\omega }}{+}{1}{,}{3}{,}{{\mathbf{\omega }}}^{{2}}{+}{1}{,}{3}{,}{{\mathbf{\omega }}}^{{2}}{+}{1}{,}{3}{,}{{\mathbf{\omega }}}^{{2}}{+}{1}{,}{4}{,}{{\mathbf{\omega }}}^{{\mathbf{\omega }}}{+}{1}{,}{5}\right]$ (3)

Display the result as a product, and verify the answer.

 > $\mathrm{Factor}\left(a,\mathrm{output}=\mathrm{inert}\right)$
 ${{\mathbf{\omega }}}^{{2}}{\mathbf{\cdot }}{2}{\mathbf{\cdot }}\left({\mathbf{\omega }}{+}{1}\right){\mathbf{\cdot }}{3}{\mathbf{\cdot }}\left({{\mathbf{\omega }}}^{{2}}{+}{1}\right){\mathbf{\cdot }}{3}{\mathbf{\cdot }}\left({{\mathbf{\omega }}}^{{2}}{+}{1}\right){\mathbf{\cdot }}{3}{\mathbf{\cdot }}\left({{\mathbf{\omega }}}^{{2}}{+}{1}\right){\mathbf{\cdot }}{4}{\mathbf{\cdot }}\left({{\mathbf{\omega }}}^{{\mathbf{\omega }}}{+}{1}\right){\mathbf{\cdot }}{5}$ (4)
 > $\mathrm{value}\left(\right)$
 ${{\mathbf{\omega }}}^{{\mathbf{\omega }}}{\cdot }{5}{+}{{\mathbf{\omega }}}^{{9}}{\cdot }{4}{+}{{\mathbf{\omega }}}^{{7}}{\cdot }{3}{+}{{\mathbf{\omega }}}^{{5}}{\cdot }{3}{+}{{\mathbf{\omega }}}^{{3}}{\cdot }{3}{+}{{\mathbf{\omega }}}^{{2}}{\cdot }{2}$ (5)

Other output forms. Note the grouping of similar factors.

 > $\mathrm{Factor}\left(a,\mathrm{output}=\mathrm{inert},\mathrm{form}=\mathrm{monic}\right)$
 ${{\mathbf{\omega }}}^{{2}}{\mathbf{\cdot }}\left({\mathbf{\omega }}{+}{2}\right){\mathbf{\cdot }}{\left({{\mathbf{\omega }}}^{{2}}{+}{3}\right)}^{{3}}{\mathbf{\cdot }}\left({{\mathbf{\omega }}}^{{\mathbf{\omega }}}{+}{4}\right){\mathbf{\cdot }}{5}$ (6)
 > $\mathrm{Factor}\left(a,\mathrm{output}=\mathrm{inert},\mathrm{form}=\mathrm{rmonic}\right)$
 $\left({{\mathbf{\omega }}}^{{2}}{\cdot }{2}\right){\mathbf{\cdot }}\left({\mathbf{\omega }}{\cdot }{3}{+}{1}\right){\mathbf{\cdot }}{\left({{\mathbf{\omega }}}^{{2}}{\cdot }{3}{+}{1}\right)}^{{2}}{\mathbf{\cdot }}\left({{\mathbf{\omega }}}^{{2}}{\cdot }{4}{+}{1}\right){\mathbf{\cdot }}\left({{\mathbf{\omega }}}^{{\mathbf{\omega }}}{\cdot }{5}{+}{1}\right)$ (7)

Just the bare data of the full factored normal form, and the original data of the Cantor normal form, for comparison.

 > $\mathrm{Factor}\left(a,\mathrm{form}=\mathrm{pairs}\right)$
 $\left[\left[{2}{,}{2}\right]{,}\left[{1}{,}{3}\right]{,}\left[{2}{,}{3}\right]{,}\left[{2}{,}{3}\right]{,}\left[{2}{,}{4}\right]{,}\left[{\mathbf{\omega }}{,}{5}\right]\right]$ (8)
 > $\mathrm{op}\left(a\right)$
 $\left[\left[{\mathbf{\omega }}{,}{5}\right]{,}\left[{9}{,}{4}\right]{,}\left[{7}{,}{3}\right]{,}\left[{5}{,}{3}\right]{,}\left[{3}{,}{3}\right]{,}\left[{2}{,}{2}\right]\right]$ (9)

Parametric examples.

 > $\mathrm{Factor}\left(a+x\right)$
 > $\mathrm{Factor}\left(a+x+7,\mathrm{form}=\mathrm{rmonic}\right)$
 $\left[{x}{+}{7}{,}{{\mathbf{\omega }}}^{{2}}{\cdot }{2}{+}{1}{,}{\mathbf{\omega }}{\cdot }{3}{+}{1}{,}{{\mathbf{\omega }}}^{{2}}{\cdot }{3}{+}{1}{,}{{\mathbf{\omega }}}^{{2}}{\cdot }{3}{+}{1}{,}{{\mathbf{\omega }}}^{{2}}{\cdot }{4}{+}{1}{,}{{\mathbf{\omega }}}^{{\mathbf{\omega }}}{\cdot }{5}{+}{1}\right]$ (10)
 > $\mathrm{Mult}\left(\mathrm{op}\left(\right)\right)$
 ${{\mathbf{\omega }}}^{{\mathbf{\omega }}}{\cdot }{5}{+}{{\mathbf{\omega }}}^{{9}}{\cdot }{4}{+}{{\mathbf{\omega }}}^{{7}}{\cdot }{3}{+}{{\mathbf{\omega }}}^{{5}}{\cdot }{3}{+}{{\mathbf{\omega }}}^{{3}}{\cdot }{3}{+}{{\mathbf{\omega }}}^{{2}}{\cdot }{2}{+}\left({x}{+}{7}\right)$ (11) Compatibility

 • The Ordinals[Factor] command was introduced in Maple 2015.