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Prime




Mathematica Notation

Traditional Notation









Number Theory Functions > Prime[n] > Other identities > Congruence properties





http://functions.wolfram.com/13.03.32.0003.01









  


  








Input Form





Mod[Binomial[2 Prime[k] - 1, Prime[k] - 1], Prime[k]^3] == 1 /; k > 2








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Mod", "[", RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", RowBox[List["Prime", "[", "k", "]"]]]], "-", "1"]], ",", RowBox[List[RowBox[List["Prime", "[", "k", "]"]], "-", "1"]]]], "]"]], ",", SuperscriptBox[RowBox[List["Prime", "[", "k", "]"]], "3"]]], "]"]], "\[Equal]", "1"]], "/;", RowBox[List["k", ">", "2"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <msup> <mrow> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <ci> Binomial </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Prime </ci> <ci> FE`Conversion`Private`k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <ci> Prime </ci> <ci> FE`Conversion`Private`k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Prime </ci> <ci> FE`Conversion`Private`k </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </annotation-xml> </semantics> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> &gt; </mo> <mn> 2 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <rem /> <apply> <ci> Binomial </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Prime </ci> <ci> FE`Conversion`Private`k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <ci> Prime </ci> <ci> FE`Conversion`Private`k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Prime </ci> <ci> FE`Conversion`Private`k </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Mod", "[", RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", RowBox[List["Prime", "[", "k", "]"]]]], "-", "1"]], ",", RowBox[List[RowBox[List["Prime", "[", "k", "]"]], "-", "1"]]]], "]"]], ",", SuperscriptBox[RowBox[List["Prime", "[", "k", "]"]], "3"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["1", "/;", RowBox[List["k", ">", "2"]]]]]]]]


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





2001-10-29