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JacobiSymbol






Mathematica Notation

Traditional Notation









Number Theory Functions > JacobiSymbol[n,m] > Series representations > Other series representations





http://functions.wolfram.com/13.08.06.0001.01









  


  










Input Form





JacobiSymbol[n, p] == (1/(I^((p - 1)/2)^2 Sqrt[p])) (1 + Sum[KroneckerDelta[GCD[k, p] - 1] Exp[(2 n Pi I k^2)/p], {k, 0, p}]) /; GCD[n, p] == 1 && Element[p, Primes] && p > 3 && Element[n - 3, Integers] && n - 3 > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiSymbol", "[", RowBox[List["n", ",", "p"]], "]"]], " ", "\[Equal]", "\[IndentingNewLine]", RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox["\[ImaginaryI]", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["p", "-", "1"]], "2"], ")"]], "2"]], " ", SqrtBox["p"]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "p"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["GCD", "[", RowBox[List["k", ",", "p"]], "]"]], "-", "1"]], "]"]], RowBox[List["Exp", "[", FractionBox[RowBox[List["2", " ", "n", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", SuperscriptBox["k", "2"]]], "p"], "]"]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["GCD", "[", RowBox[List["n", ",", "p"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List["p", "\[Element]", "Primes"]], "\[And]", RowBox[List["p", ">", "3"]], "\[And]", RowBox[List[RowBox[List["n", "-", "3"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["n", "-", "3"]], ">", "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> <semantics> <mrow> <mo> ( </mo> <mfrac> <mi> n </mi> <mi> p </mi> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;n&quot;, &quot;p&quot;], &quot;)&quot;]], JacobiSymbol, Rule[Editable, False]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> &#8520; </mi> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msup> <mo> &#8290; </mo> <msqrt> <mi> p </mi> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#8290; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> k </mi> <mn> 2 </mn> </msup> </mrow> <mi> p </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &#8712; </mo> <semantics> <mi> &#8473; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalP]&quot;, Function[Primes]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &gt; </mo> <mn> 3 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <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> JacobiSymbol </ci> <ci> n </ci> <ci> p </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <imaginaryi /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> p </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> p </ci> </uplimit> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <apply> <gcd /> <ci> k </ci> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <exp /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> <pi /> <imaginaryi /> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <gcd /> <ci> n </ci> <ci> p </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <ci> p </ci> <primes /> </apply> <apply> <gt /> <ci> p </ci> <cn type='integer'> 3 </cn> </apply> <apply> <in /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -3 </cn> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiSymbol", "[", RowBox[List["n_", ",", "p_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "p"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["GCD", "[", RowBox[List["k", ",", "p"]], "]"]], "-", "1"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "n", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", SuperscriptBox["k", "2"]]], "p"]]]]]]]], RowBox[List[SuperscriptBox["\[ImaginaryI]", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["p", "-", "1"]], "2"], ")"]], "2"]], " ", SqrtBox["p"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["GCD", "[", RowBox[List["n", ",", "p"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List["p", "\[Element]", "Primes"]], "&&", RowBox[List["p", ">", "3"]], "&&", RowBox[List[RowBox[List["n", "-", "3"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["n", "-", "3"]], ">", "0"]]]]]]]]]]










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





2001-10-29