Product[JacobiSymbol[Subscript[p, k], m], {k, 1, r}] == JacobiSymbol[n, m] /; n == Product[Subscript[p, k]^Subscript[n, k], {k, 1, r}] && Element[Subscript[p, k], Primes] && GCD[n, m] == 1 && Element[(n - 3)/2, Integers] && (n - 3)/2 >= 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "r"], RowBox[List["JacobiSymbol", "[", RowBox[List[SubscriptBox["p", "k"], ",", "m"]], "]"]]]], "\[Equal]", RowBox[List["JacobiSymbol", "[", RowBox[List["n", ",", "m"]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "r"], SuperscriptBox[SubscriptBox["p", "k"], SubscriptBox["n", "k"]]]]]], "\[And]", RowBox[List[SubscriptBox["p", "k"], "\[Element]", "Primes"]], "\[And]", RowBox[List[RowBox[List["GCD", "[", RowBox[List["n", ",", "m"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List[FractionBox[RowBox[List["n", "-", "3"]], "2"], "\[Element]", "Integers"]], " ", "\[And]", RowBox[List[FractionBox[RowBox[List["n", "-", "3"]], "2"], "\[GreaterEqual]", "0"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </munderover> <semantics> <mrow> <mo> ( </mo> <mfrac> <msub> <mi> p </mi> <mi> k </mi> </msub> <mi> m </mi> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, FractionBox[SubscriptBox[&quot;p&quot;, &quot;k&quot;], &quot;m&quot;], &quot;)&quot;]], JacobiSymbol, Rule[Editable, False]] </annotation> </semantics> </mrow> <mo> &#10869; </mo> <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> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </munderover> <msubsup> <mi> p </mi> <mi> k </mi> <msub> <mi> n </mi> <mi> k </mi> </msub> </msubsup> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8473; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalP]&quot;, Function[Primes]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> r </ci> </uplimit> <apply> <ci> JacobiSymbol </ci> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <ci> m </ci> </apply> </apply> <apply> <ci> JacobiSymbol </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> n </ci> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> r </ci> </uplimit> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <primes /> </apply> <apply> <eq /> <apply> <gcd /> <ci> n </ci> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -3 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "r_"], RowBox[List["JacobiSymbol", "[", RowBox[List[SubscriptBox["p_", "k"], ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["JacobiSymbol", "[", RowBox[List["n", ",", "m"]], "]"]], "/;", RowBox[List[RowBox[List["n", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "r"], SubsuperscriptBox["p", "k", SubscriptBox["n", "k"]]]]]], "&&", RowBox[List[SubscriptBox["p", "k"], "\[Element]", "Primes"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["n", ",", "m"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[FractionBox[RowBox[List["n", "-", "3"]], "2"], "\[Element]", "Integers"]], "&&", RowBox[List[FractionBox[RowBox[List["n", "-", "3"]], "2"], "\[GreaterEqual]", "0"]]]]]]]]]]

2001-10-29