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JacobiSymbol




Mathematica Notation

Traditional Notation









Number Theory Functions > JacobiSymbol[n,m] > Transformations > Products, sums, and powers of the direct function > Products of the direct function





http://functions.wolfram.com/13.08.16.0004.01









  


  








Input Form





JacobiSymbol[m, n] JacobiSymbol[n, m] == (-1)^((m - 1) ((n - 1)/4)) /; GCD[m, n] == 1 && Element[(n - 1)/2, Integers] && (n - 1)/2 >= 0 && Element[(m - 1)/2, Integers] && (m - 1)/2 >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["JacobiSymbol", "[", RowBox[List["m", ",", "n"]], "]"]], RowBox[List["JacobiSymbol", "[", RowBox[List["n", ",", "m"]], "]"]]]], "\[Equal]", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "/", "4"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[FractionBox[RowBox[List["m", "-", "1"]], "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List[FractionBox[RowBox[List["m", "-", "1"]], "2"], "\[GreaterEqual]", "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> <mrow> <mo> ( </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;m&quot;, &quot;n&quot;], &quot;)&quot;]], JacobiSymbol, Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mfrac> <mi> n </mi> <mi> m </mi> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;n&quot;, &quot;m&quot;], &quot;)&quot;]], JacobiSymbol, Rule[Editable, False]] </annotation> </semantics> </mrow> <mo> &#10869; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mfrac> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> JacobiSymbol </ci> <ci> m </ci> <ci> n </ci> </apply> <apply> <ci> JacobiSymbol </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <gcd /> <ci> m </ci> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8469; </ci> </apply> <apply> <in /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["JacobiSymbol", "[", RowBox[List["m_", ",", "n_"]], "]"]], " ", RowBox[List["JacobiSymbol", "[", RowBox[List["n_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "\[Element]", "Integers"]], "&&", RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "\[GreaterEqual]", "0"]], "&&", RowBox[List[FractionBox[RowBox[List["m", "-", "1"]], "2"], "\[Element]", "Integers"]], "&&", RowBox[List[FractionBox[RowBox[List["m", "-", "1"]], "2"], "\[GreaterEqual]", "0"]]]]]]]]]]


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





2001-10-29