EulerPhi[n^k] == EulerPhi[n] n^(k - 1) /; Element[n, Integers] && n > 0 && Element[k, Integers] && k > 0

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <msup> <mi> n </mi> <mi> k </mi> </msup> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> n </mi> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EulerPhi </ci> <apply> <power /> <ci> n </ci> <ci> k </ci> </apply> </apply> <apply> <times /> <apply> <ci> EulerPhi </ci> <ci> n </ci> </apply> <apply> <power /> <ci> n </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> k </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EulerPhi", "[", SuperscriptBox["n_", "k"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["EulerPhi", "[", "n", "]"]], " ", SuperscriptBox["n", RowBox[List["k", "-", "1"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]]]]]]]]]]

2001-10-29