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JacobiSymbol






Mathematica Notation

Traditional Notation









Number Theory Functions > JacobiSymbol[n,m] > Identities > Functional identities





http://functions.wolfram.com/13.08.17.0008.01









  


  










Input Form





JacobiSymbol[Subscript[a, 1], Subscript[c, 1]] JacobiSymbol[Subscript[a, 2], Subscript[c, 2]] JacobiSymbol[Subscript[a, 3], Subscript[c, 3]] == (-1)^(((Subscript[c, 1] - 1) (Subscript[c, 2] - 1) + (Subscript[c, 1] - 1) (Subscript[c, 3] - 1) + (Subscript[c, 2] - 1) (Subscript[c, 3] - 1))/4) /; {{Subscript[a, 1], Subscript[b, 1]}, {Subscript[c, 1], Subscript[d, 1]}} . {{Subscript[a, 2], Subscript[b, 2]}, {Subscript[c, 2], Subscript[d, 2]}} . {{Subscript[a, 3], Subscript[b, 3]}, {Subscript[c, 3], Subscript[d, 3]}} == {{-1, 0}, {0, -1}} && Subscript[a, 1] Subscript[d, 1] - Subscript[b, 1] Subscript[c, 1] == 1 && Subscript[a, 2] Subscript[d, 2] - Subscript[b, 2] Subscript[c, 2] == 1 && Subscript[a, 3] Subscript[d, 3] - Subscript[b, 3] Subscript[c, 3] == 1 && Element[{Subscript[a, 1], Subscript[b, 1], Subscript[d, 1], Subscript[a, 2], Subscript[b, 2], Subscript[d, 2], Subscript[a, 3], Subscript[b, 3], Subscript[d, 3]}, Integers] && Element[{(Subscript[c, 1] + 1)/2, (Subscript[c, 2] + 1)/2, (Subscript[c, 3] + 1)/2}, Integers] && Subscript[c, 1] > 0 && Subscript[c, 2] > 0 && Subscript[c, 3] > 0










Standard Form





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MathML Form







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</msub> <mo> &#8290; </mo> <msub> <mi> c </mi> <mn> 1 </mn> </msub> </mrow> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> d </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> c </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> d </mi> <mn> 3 </mn> </msub> </mrow> <mo> - </mo> <mrow> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> c </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> d </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> 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&#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> JacobiSymbol </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> JacobiSymbol </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> JacobiSymbol </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> 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</apply> <apply> <ci> Subscript </ci> <ci> d </ci> <cn type='integer'> 3 </cn> </apply> </list> </list> </apply> <list> <list> <cn type='integer'> -1 </cn> <cn type='integer'> 0 </cn> </list> <list> <cn type='integer'> -1 </cn> <cn type='integer'> 0 </cn> </list> </list> </apply> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> d </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> d </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> d </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> d </ci> <cn type='integer'> 3 </cn> </apply> </list> <integers /> </apply> <apply> <in /> <list> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





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