JacobiSymbol[n, m] == Product[JacobiSymbol[n, Subscript[p, k]], {k, 1, j}] /; Element[(m - 1)/2, Integers] && (m - 1)/2 >= 0 && FactorInteger[m] == {{Subscript[p, 1], 1}, {Subscript[p, 2], 1}, \[Ellipsis], {Subscript[p, j], 1}} && (JacobiSymbol[n, p] == (1 - KroneckerDelta[n/p - Floor[n/p], 0]) (2 Sign[Sum[KroneckerDelta[Mod[j^2, p], Mod[n, p]], {j, 1, p}]] - 1) /; Element[p, Primes])

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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> 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> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> j </mi> </munderover> <semantics> <mrow> <mo> ( </mo> <mfrac> <mi> n </mi> <msub> <mi> p </mi> <mi> k </mi> </msub> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;n&quot;, SubscriptBox[&quot;p&quot;, &quot;k&quot;]], &quot;)&quot;]], JacobiSymbol, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> factors </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> p </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> p </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> p </mi> <mi> j </mi> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <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> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mfrac> <mi> n </mi> <mi> p </mi> </mfrac> <mo> - </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mi> n </mi> <mi> p </mi> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <semantics> <mrow> <msup> <mi> j </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <power /> <ci> FE`Conversion`Private`j </ci> <cn type='integer'> 2 </cn> </apply> <ci> FE`Conversion`Private`p </ci> </apply> </annotation-xml> </semantics> <mo> , </mo> <semantics> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> FE`Conversion`Private`n </ci> <ci> FE`Conversion`Private`p </ci> </apply> </annotation-xml> </semantics> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> p </mi> <mo> &#8712; </mo> <semantics> <mi> &#8473; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalP]&quot;, Function[Primes]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> JacobiSymbol </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> j </ci> </uplimit> <apply> <ci> JacobiSymbol </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <and /> <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> <integers /> </apply> <apply> <eq /> <apply> <ci> factors </ci> <ci> m </ci> </apply> <list> <list> <apply> <ci> Subscript </ci> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </list> <list> <apply> <ci> Subscript </ci> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </list> <ci> &#8230; </ci> <list> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </list> </list> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> JacobiSymbol </ci> <ci> n </ci> <ci> p </ci> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Sign </ci> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> p </ci> </uplimit> <apply> <ci> KroneckerDelta </ci> <apply> <rem /> <apply> <power /> <ci> FE`Conversion`Private`j </ci> <cn type='integer'> 2 </cn> </apply> <ci> FE`Conversion`Private`p </ci> </apply> <apply> <rem /> <ci> FE`Conversion`Private`n </ci> <ci> FE`Conversion`Private`p </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> p </ci> <primes /> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29