Mod[Binomial[Sum[Subscript[m, k] p^k, {k, 0, l}], Sum[Subscript[n, k] p^k, {k, 0, l}]], p] == Mod[Product[Binomial[Subscript[m, k], Subscript[n, k]], {k, 0, l}], p] /; Element[p, Primes] && Element[Subscript[m, k], Integers] && Subscript[m, k] >= 0 && Element[Subscript[n, k], Integers] && Subscript[n, k] >= 0 && Subscript[m, k] < p && Subscript[n, k] < p

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<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> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> l </mi> </munderover> <mrow> <msub> <mi> m </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msup> <mi> p </mi> <mi> k </mi> </msup> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> l </mi> </munderover> <mrow> <msub> <mi> n </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msup> <mi> p </mi> <mi> k </mi> </msup> </mrow> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <ci> Binomial </ci> <apply> <sum /> <bvar> <ci> FE`Conversion`Private`k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> FE`Conversion`Private`l </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> FE`Conversion`Private`m </ci> <ci> FE`Conversion`Private`k </ci> </apply> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <ci> FE`Conversion`Private`k </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> FE`Conversion`Private`k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> FE`Conversion`Private`l </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> FE`Conversion`Private`n </ci> <ci> FE`Conversion`Private`k </ci> </apply> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <ci> FE`Conversion`Private`k </ci> </apply> </apply> </apply> </apply> <ci> FE`Conversion`Private`p </ci> </apply> </annotation-xml> </semantics> <mo> &#10869; </mo> <semantics> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> l </mi> </munderover> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <msub> <mi> m </mi> <mi> k </mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> n </mi> <mi> k </mi> </msub> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <product /> <bvar> <ci> FE`Conversion`Private`k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> FE`Conversion`Private`l </ci> </uplimit> <apply> <ci> Binomial </ci> <apply> <ci> Subscript </ci> <ci> FE`Conversion`Private`m </ci> <ci> FE`Conversion`Private`k </ci> </apply> <apply> <ci> Subscript </ci> <ci> FE`Conversion`Private`n </ci> <ci> FE`Conversion`Private`k </ci> </apply> </apply> </apply> <ci> FE`Conversion`Private`p </ci> </apply> </annotation-xml> </semantics> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> &#8712; </mo> <semantics> <mi> &#8473; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalP]&quot;, Function[Primes]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> m </mi> <mi> k </mi> </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> n </mi> <mi> k </mi> </msub> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> m </mi> <mi> k </mi> </msub> <mo> &lt; </mo> <mi> p </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mi> k </mi> </msub> <mo> &lt; </mo> <mi> p </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <rem /> <apply> <ci> Binomial </ci> <apply> <sum /> <bvar> <ci> FE`Conversion`Private`k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> FE`Conversion`Private`l </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> FE`Conversion`Private`m </ci> <ci> FE`Conversion`Private`k </ci> </apply> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <ci> FE`Conversion`Private`k </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> FE`Conversion`Private`k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> FE`Conversion`Private`l </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> FE`Conversion`Private`n </ci> <ci> FE`Conversion`Private`k </ci> </apply> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <ci> FE`Conversion`Private`k </ci> </apply> </apply> </apply> </apply> <ci> FE`Conversion`Private`p </ci> </apply> <apply> <rem /> <apply> <product /> <bvar> <ci> FE`Conversion`Private`k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> FE`Conversion`Private`l </ci> </uplimit> <apply> <ci> Binomial </ci> <apply> <ci> Subscript </ci> <ci> FE`Conversion`Private`m </ci> <ci> FE`Conversion`Private`k </ci> </apply> <apply> <ci> Subscript </ci> <ci> FE`Conversion`Private`n </ci> <ci> FE`Conversion`Private`k </ci> </apply> </apply> </apply> <ci> FE`Conversion`Private`p </ci> </apply> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <primes /> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> m </ci> <ci> k </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> <ci> &#8469; </ci> </apply> <apply> <lt /> <apply> <ci> Subscript </ci> <ci> m </ci> <ci> k </ci> </apply> <ci> p </ci> </apply> <apply> <lt /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> <ci> p </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29