JacobiSymbol[n, p] == Sign[p/2 - Mod[n^((p - 1)/2), p]] /; Element[p, Primes] && Element[(n - 1)/2, Integers] && (n - 1)/2 >= 0 && GCD[n, p] == 1

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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> <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> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> p </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <semantics> <mrow> <msup> <mi> n </mi> <mfrac> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <power /> <ci> $CellContext`n </ci> <apply> <times /> <apply> <plus /> <ci> FE`Conversion`Private`p </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> FE`Conversion`Private`p </ci> </apply> </annotation-xml> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <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> <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> <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> </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> <ci> Sign </ci> <apply> <plus /> <apply> <times /> <ci> p </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <rem /> <apply> <power /> <ci> $CellContext`n </ci> <apply> <times /> <apply> <plus /> <ci> FE`Conversion`Private`p </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> FE`Conversion`Private`p </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <primes /> </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> <eq /> <apply> <gcd /> <ci> n </ci> <ci> p </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiSymbol", "[", RowBox[List["n_", ",", "p_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Sign", "[", RowBox[List[FractionBox["p", "2"], "-", RowBox[List["Mod", "[", RowBox[List[SuperscriptBox["n", FractionBox[RowBox[List["p", "-", "1"]], "2"]], ",", "p"]], "]"]]]], "]"]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Primes"]], "&&", RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "\[Element]", "Integers"]], "&&", RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["n", ",", "p"]], "]"]], "\[Equal]", "1"]]]]]]]]]]

2001-10-29