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CarmichaelLambda




Mathematica Notation

Traditional Notation









Number Theory Functions > CarmichaelLambda[n] > Primary definition





http://functions.wolfram.com/13.09.02.0001.01









  


  








Input Form





CarmichaelLambda[n] == \[Lambda] /; ForAll[m, GCD[n, m] == 1, Mod[m^\[Lambda], n] == 1] && ForAll[\[CapitalLambda], !(\[CapitalLambda] < \[Lambda] && Mod[m^\[Lambda], n] == 1 && GCD[m, n] == 1)]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["CarmichaelLambda", "[", "n", "]"]], "\[Equal]", "\[Lambda]"]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[ForAll]", RowBox[List["m", ",", RowBox[List[RowBox[List["GCD", "[", RowBox[List["n", ",", "m"]], "]"]], "\[Equal]", "1"]]]]], RowBox[List[RowBox[List["Mod", "[", RowBox[List[SuperscriptBox["m", "\[Lambda]"], ",", "n"]], "]"]], "\[Equal]", "1"]]]], "&&", RowBox[List[SubscriptBox["\[ForAll]", "\[CapitalLambda]"], RowBox[List["(", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["\[CapitalLambda]", "<", "\[Lambda]"]], "&&", RowBox[List[RowBox[List["Mod", "[", RowBox[List[SuperscriptBox["m", "\[Lambda]"], ",", "n"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]], "\[Equal]", "1"]]]], ")"]]]], ")"]]]]]]]]]]








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> &#955; </mi> <annotation-xml encoding='MathML-Content'> <ci> CarmichaelLambda </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mi> &#955; </mi> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mo> &#8704; </mo> <mrow> <mi> m </mi> <mo> , </mo> <mtext> </mtext> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> </mrow> </msub> <mrow> <semantics> <mrow> <msup> <mi> m </mi> <mi> &#955; </mi> </msup> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mi> n </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <power /> <ci> $CellContext`m </ci> <ci> $CellContext`&#955; </ci> </apply> <ci> $CellContext`n </ci> </apply> </annotation-xml> </semantics> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mo> &#8704; </mo> <mi> &#923; </mi> </msub> <mrow> <mo> ( </mo> <mrow> <mo> &#172; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#923; </mi> <mo> &lt; </mo> <mi> &#955; </mi> </mrow> <mo> &#8743; </mo> <mrow> <semantics> <mrow> <msup> <mi> m </mi> <mi> &#955; </mi> </msup> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mi> n </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <power /> <ci> $CellContext`m </ci> <ci> $CellContext`&#955; </ci> </apply> <ci> $CellContext`n </ci> </apply> </annotation-xml> </semantics> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> CarmichaelLambda </ci> <ci> n </ci> </apply> <ci> &#955; </ci> </apply> <apply> <and /> <apply> <forall /> <bvar> <ci> m </ci> </bvar> <bvar> <apply> <eq /> <apply> <gcd /> <ci> n </ci> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> </bvar> <apply> <eq /> <apply> <rem /> <apply> <power /> <ci> $CellContext`m </ci> <ci> $CellContext`&#955; </ci> </apply> <ci> $CellContext`n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <forall /> <bvar> <ci> &#923; </ci> </bvar> <apply> <not /> <apply> <and /> <apply> <lt /> <ci> &#923; </ci> <ci> &#955; </ci> </apply> <apply> <eq /> <apply> <rem /> <apply> <power /> <ci> $CellContext`m </ci> <ci> $CellContext`&#955; </ci> </apply> <ci> $CellContext`n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <gcd /> <ci> n </ci> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["CarmichaelLambda", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List["\[Lambda]", "/;", RowBox[List[RowBox[List[SubscriptBox["\[ForAll]", RowBox[List["m", ",", RowBox[List[RowBox[List["GCD", "[", RowBox[List["n", ",", "m"]], "]"]], "\[Equal]", "1"]]]]], RowBox[List[RowBox[List["Mod", "[", RowBox[List[SuperscriptBox["m", "\[Lambda]"], ",", "n"]], "]"]], "\[Equal]", "1"]]]], "&&", RowBox[List[SubscriptBox["\[ForAll]", "\[CapitalLambda]"], RowBox[List["(", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["\[CapitalLambda]", "<", "\[Lambda]"]], "&&", RowBox[List[RowBox[List["Mod", "[", RowBox[List[SuperscriptBox["m", "\[Lambda]"], ",", "n"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]], "\[Equal]", "1"]]]], ")"]]]], ")"]]]]]]]]]]]]


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





2001-10-29