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PartitionsP






Mathematica Notation

Traditional Notation









Integer Functions > PartitionsP[n] > Other identities > Congruence properties





http://functions.wolfram.com/04.16.32.0004.01









  


  










Input Form





Mod[PartitionsP[n], 5^Subscript[k, 1] 7^(Subscript[k, 2] + 1) 11^Subscript[k, 3]] == 0 /; Mod[24 n, 5^Subscript[k, 1] 7^Subscript[k, 2] 11^Subscript[k, 3]] == 1 && Element[Subscript[k, 1], Integers] && Subscript[k, 1] >= 0 && Element[Subscript[k, 2], Integers] && Subscript[k, 2] >= 0 && Element[Subscript[k, 3], Integers] && Subscript[k, 3] >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Mod", "[", RowBox[List[RowBox[List["PartitionsP", "[", "n", "]"]], ",", RowBox[List[SuperscriptBox["5", SubscriptBox["k", "1"]], " ", SuperscriptBox["7", RowBox[List[SubscriptBox["k", "2"], "+", "1"]]], " ", SuperscriptBox["11", SubscriptBox["k", "3"]]]]]], "]"]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List[RowBox[List["Mod", "[", RowBox[List[RowBox[List["24", " ", "n"]], ",", RowBox[List[SuperscriptBox["5", SubscriptBox["k", "1"]], " ", SuperscriptBox["7", SubscriptBox["k", "2"]], " ", SuperscriptBox["11", SubscriptBox["k", "3"]]]]]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["k", "1"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["k", "1"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["k", "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["k", "2"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["k", "3"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["k", "3"], "\[GreaterEqual]", "0"]]]]]]]]










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> <mi> p </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mn> 5 </mn> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </msup> <mo> &#8290; </mo> <msup> <mn> 7 </mn> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 11 </mn> <msub> <mi> k </mi> <mn> 3 </mn> </msub> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <ci> PartitionsP </ci> <ci> $CellContext`n </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 5 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 7 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> $CellContext`k </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 11 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`k </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mn> 5 </mn> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </msup> <mo> &#8290; </mo> <msup> <mn> 7 </mn> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </msup> <mo> &#8290; </mo> <msup> <mn> 11 </mn> <msub> <mi> k </mi> <mn> 3 </mn> </msub> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <times /> <cn type='integer'> 24 </cn> <ci> $CellContext`n </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 5 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 7 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 11 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`k </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> k </mi> <mn> 3 </mn> </msub> <mo> &#8712; </mo> <semantics> <mi> &#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> <rem /> <apply> <ci> PartitionsP </ci> <ci> $CellContext`n </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 5 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 7 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> $CellContext`k </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 11 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`k </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <eq /> <apply> <rem /> <apply> <times /> <cn type='integer'> 24 </cn> <ci> $CellContext`n </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 5 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 7 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 11 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`k </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <integers /> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <integers /> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Mod", "[", RowBox[List[RowBox[List["PartitionsP", "[", "n_", "]"]], ",", RowBox[List[SuperscriptBox["5", SubscriptBox["k_", "1"]], " ", SuperscriptBox["7", RowBox[List[SubscriptBox["k_", "2"], "+", "1"]]], " ", SuperscriptBox["11", SubscriptBox["k_", "3"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List[RowBox[List["Mod", "[", RowBox[List[RowBox[List["24", " ", "n"]], ",", RowBox[List[SuperscriptBox["5", SubscriptBox["k", "1"]], " ", SuperscriptBox["7", SubscriptBox["k", "2"]], " ", SuperscriptBox["11", SubscriptBox["k", "3"]]]]]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["k", "1"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["k", "1"], "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["k", "2"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["k", "2"], "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["k", "3"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["k", "3"], "\[GreaterEqual]", "0"]]]]]]]]]]










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





2001-10-29





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