Euler totient function: Representations through equivalent functions (formula 13.06.27.0002)

EulerPhi

$Mathematica Notation$

Number Theory Functions

EulerPhi[n]

Representations through equivalent functions

With related functions

http://functions.wolfram.com/13.06.27.0002.01

Input Form

EulerPhi[p^n] == CarmichaelLambda[p^n] /; Element[p, Primes] && p > 2 && Element[n, Integers] && n > 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EulerPhi", "[", SuperscriptBox["p", "n"], "]"]], "\[Equal]", RowBox[List["CarmichaelLambda", "[", " ", SuperscriptBox["p", "n"], "]"]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Primes"]], "\[And]", RowBox[List["p", ">", "2"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

MathML Form

<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> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", EulerPhi] </annotation> </semantics> <mo> ( </mo> <msup> <mi> p </mi> <mi> n </mi> </msup> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <semantics> <mi> λ </mi> <annotation-xml encoding='MathML-Content'> <ci> CarmichaelLambda </ci> </annotation-xml> </semantics> <mo> ( </mo> <msup> <mi> p </mi> <mi> n </mi> </msup> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> ∈ </mo> <semantics> <mi> ℙ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalP]", Function[Primes]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> p </mi> <mo> > </mo> <mn> 2 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </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> p </ci> <ci> n </ci> </apply> </apply> <apply> <ci> CarmichaelLambda </ci> <apply> <power /> <ci> p </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <primes /> </apply> <apply> <gt /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EulerPhi", "[", SuperscriptBox["p_", "n_"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["CarmichaelLambda", "[", SuperscriptBox["p", "n"], "]"]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Primes"]], "&&", RowBox[List["p", ">", "2"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

Date Added to functions.wolfram.com (modification date)

2001-10-29