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

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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> <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>

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2001-10-29