Mod[((1 - p^(n - 1))/n) BernoulliB[n], p^e] == (r^n/r^(n - 1)) ((r^EulerPhi[p^e] - 1)/p^e) - Sum[r^(k (n - 1)) Floor[r^k/p^e], {k, 1, EulerPhi[p^e]}] /; Element[p, Primes] && p >= 5 && Element[n, Integers] && n > 0 && NotElement[(p - 1)/n, Integers] && Element[e, Integers] && e >= 1 && Sort[Mod[r^Range[0, p^e - 2], p^e]] == Range[p^e - 1]

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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> <mrow> <mfrac> <mrow> <msup> <mi> r </mi> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> r </mi> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <msup> <mi> p </mi> <mi> e </mi> </msup> <mo> ) </mo> </mrow> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> r </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> p </mi> <mi> e </mi> </msup> </mrow> </mfrac> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <msup> <mi> p </mi> <mi> e </mi> </msup> <mo> ) </mo> </mrow> </munderover> <mrow> <msup> <mi> r </mi> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <msup> <mi> r </mi> <mi> k </mi> </msup> <msup> <mi> p </mi> <mi> e </mi> </msup> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </mrow> </mrow> </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> <mi> p </mi> <mo> &#8805; </mo> <mn> 5 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mfrac> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> e </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> e </mi> <mo> &#8805; </mo> <mn> 1 </mn> </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> <plus /> <apply> <times /> <apply> <power /> <ci> r </ci> <ci> n </ci> </apply> <apply> <plus /> <apply> <power /> <ci> r </ci> <apply> <ci> EulerPhi </ci> <apply> <power /> <ci> p </ci> <ci> e </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> r </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> p </ci> <ci> e </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <ci> EulerPhi </ci> <apply> <power /> <ci> p </ci> <ci> e </ci> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <ci> r </ci> <apply> <times /> <ci> k </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <power /> <ci> r </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <power /> <ci> p </ci> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </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> <notin /> <apply> <times /> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> <apply> <in /> <ci> e </ci> <integers /> </apply> <apply> <geq /> <ci> e </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18