Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Mod




Mathematica Notation

Traditional Notation









Integer Functions > Mod[m,n] > Specific values > Specialized values





http://functions.wolfram.com/04.06.03.0012.01









  


  








Input Form





Mod[Abs[BernoulliB[2 n]], 1] == KroneckerDelta[Mod[(n + 1)/2, 1], 0] + (-1)^n Mod[1/2 + Sum[CharacteristicFunction[(2 n)/(k - 1), Integers] CharacteristicFunction[k, Primes] (1/k), {k, 3, 2 n + 1}], 1]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Mod", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List["BernoulliB", "[", RowBox[List["2", "n"]], "]"]], "]"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List[FractionBox[RowBox[List["n", "+", "1"]], "2"], ",", "1"]], "]"]], ",", "0"]], "]"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["Mod", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "3"]], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]], RowBox[List[RowBox[List["CharacteristicFunction", "[", RowBox[List[FractionBox[RowBox[List["2", "n"]], RowBox[List["k", "-", "1"]]], ",", "Integers"]], "]"]], RowBox[List["CharacteristicFunction", "[", RowBox[List["k", ",", "Primes"]], "]"]], FractionBox["1", "k"]]]]]]], ",", "1"]], "]"]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mrow> <mo> &#10072; </mo> <msub> <mi> B </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msub> <mo> &#10072; </mo> </mrow> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <abs /> <apply> <ci> BernoulliB </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> $CellContext`n </ci> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> <mo> &#10869; </mo> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <semantics> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <times /> <apply> <plus /> <ci> $CellContext`n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <semantics> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 3 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mi> k </mi> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8484; </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#967; </mi> <mi> &#8473; </mi> </msub> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <sum /> <bvar> <ci> $CellContext`k </ci> </bvar> <lowlimit> <cn type='integer'> 3 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> $CellContext`n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> CharacteristicFunction </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> $CellContext`n </ci> <apply> <power /> <apply> <plus /> <ci> $CellContext`k </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> <apply> <ci> CharacteristicFunction </ci> <ci> $CellContext`k </ci> <primes /> </apply> <apply> <power /> <ci> $CellContext`k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <rem /> <apply> <abs /> <apply> <ci> BernoulliB </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> $CellContext`n </ci> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> KroneckerDelta </ci> <apply> <rem /> <apply> <times /> <apply> <plus /> <ci> $CellContext`n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <rem /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <sum /> <bvar> <ci> $CellContext`k </ci> </bvar> <lowlimit> <cn type='integer'> 3 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> $CellContext`n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> CharacteristicFunction </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> $CellContext`n </ci> <apply> <power /> <apply> <plus /> <ci> $CellContext`k </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> <apply> <ci> CharacteristicFunction </ci> <ci> $CellContext`k </ci> <primes /> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> $CellContext`k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Mod", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "n_"]], "]"]], "]"]], ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List[FractionBox[RowBox[List["n", "+", "1"]], "2"], ",", "1"]], "]"]], ",", "0"]], "]"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Mod", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "3"]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], FractionBox[RowBox[List[RowBox[List["CharacteristicFunction", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "n"]], RowBox[List["k", "-", "1"]]], ",", "Integers"]], "]"]], " ", RowBox[List["CharacteristicFunction", "[", RowBox[List["k", ",", "Primes"]], "]"]]]], "k"]]]]], ",", "1"]], "]"]]]]]]]]]]


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





2001-10-29