JacobiSymbol[n, m] == JacobiSymbol[Mod[n, m], m] /; Element[(m - 1)/2, Integers] && (m - 1)/2 >= 0 && Element[n, Integers] && n >= m

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiSymbol", "[", RowBox[List["n", ",", "m"]], "]"]], "\[Equal]", RowBox[List["JacobiSymbol", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["n", ",", "m"]], "]"]], ",", "m"]], "]"]]]], "/;", RowBox[List[RowBox[List[FractionBox[RowBox[List["m", "-", "1"]], "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List[FractionBox[RowBox[List["m", "-", "1"]], "2"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "m"]]]]]]]]

<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> <semantics> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mi> m </mi> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, FractionBox[InterpretationBox[RowBox[List[&quot;n&quot;, &quot; &quot;, &quot;mod&quot;, &quot; &quot;, &quot;m&quot;]], Mod[$CellContext`n, FE`Conversion`Private`m]], &quot;m&quot;], &quot;)&quot;]], JacobiSymbol, Rule[Editable, False]] </annotation> </semantics> </mrow> <mo> /; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8805; </mo> <mi> m </mi> </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> <ci> JacobiSymbol </ci> <apply> <rem /> <ci> $CellContext`n </ci> <ci> FE`Conversion`Private`m </ci> </apply> <ci> m </ci> </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> <ci> &#8469; </ci> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <geq /> <ci> n </ci> <ci> m </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiSymbol", "[", RowBox[List["n_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["JacobiSymbol", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["n", ",", "m"]], "]"]], ",", "m"]], "]"]], "/;", RowBox[List[RowBox[List[FractionBox[RowBox[List["m", "-", "1"]], "2"], "\[Element]", "Integers"]], "&&", RowBox[List[FractionBox[RowBox[List["m", "-", "1"]], "2"], "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "m"]]]]]]]]]]

2001-10-29