Mod[((1 - p^(n - 1))/n) BernoulliB[n], p^e] == Mod[((1 - p^(m - 1))/m) BernoulliB[m], p^e] /; Element[p, Primes] && p >= 5 && Element[n, Integers] && n > 0 && Element[m, Integers] && m > 0 && Mod[n, p^e] == Mod[m, p^e]

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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> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> p </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> B </mi> <mi> n </mi> </msub> </mrow> <mi> n </mi> </mfrac> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <msup> <mi> p </mi> <mi> e </mi> </msup> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <apply> <plus /> <ci> FE`Conversion`Private`n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> BernoulliB </ci> <ci> FE`Conversion`Private`n </ci> </apply> <apply> <power /> <ci> FE`Conversion`Private`n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <ci> FE`Conversion`Private`e </ci> </apply> </apply> </annotation-xml> </semantics> <mo> &#10869; </mo> <semantics> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> p </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> B </mi> <mi> m </mi> </msub> </mrow> <mi> m </mi> </mfrac> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <msup> <mi> p </mi> <mi> e </mi> </msup> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <apply> <plus /> <ci> FE`Conversion`Private`m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> BernoulliB </ci> <ci> FE`Conversion`Private`m </ci> </apply> <apply> <power /> <ci> FE`Conversion`Private`m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <ci> FE`Conversion`Private`e </ci> </apply> </apply> </annotation-xml> </semantics> </mrow> <mo> /; 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</mo> <msup> <mi> p </mi> <mi> e </mi> </msup> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> FE`Conversion`Private`n </ci> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <ci> FE`Conversion`Private`e </ci> </apply> </apply> </annotation-xml> </semantics> <mo> &#10869; </mo> <semantics> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <msup> <mi> p </mi> <mi> e </mi> </msup> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> FE`Conversion`Private`m </ci> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <ci> FE`Conversion`Private`e </ci> </apply> </apply> </annotation-xml> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <rem /> <apply> <times /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <apply> <plus /> <ci> FE`Conversion`Private`n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> FE`Conversion`Private`n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BernoulliB </ci> <ci> FE`Conversion`Private`n </ci> </apply> </apply> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <ci> FE`Conversion`Private`e </ci> </apply> </apply> <apply> <rem /> <apply> <times /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <apply> <plus /> <ci> FE`Conversion`Private`m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> FE`Conversion`Private`m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BernoulliB </ci> <ci> FE`Conversion`Private`m </ci> </apply> </apply> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <ci> FE`Conversion`Private`e </ci> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <primes /> </apply> <apply> <geq /> <ci> p </ci> <cn type='integer'> 5 </cn> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <gt /> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <gt /> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <rem /> <ci> FE`Conversion`Private`n </ci> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <ci> FE`Conversion`Private`e </ci> </apply> </apply> <apply> <rem /> <ci> FE`Conversion`Private`m </ci> <apply> <power /> <ci> FE`Conversion`Private`p </ci> <ci> FE`Conversion`Private`e </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18