Mod[Binomial[2 p - 1, p - 1], p^3] == 1 /; Element[p, Primes] && p > 3

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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> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> p </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <msup> <mi> p </mi> <mn> 3 </mn> </msup> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <ci> Binomial </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> $CellContext`p </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> $CellContext`p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> $CellContext`p </ci> <cn type='integer'> 3 </cn> </apply> </apply> </annotation-xml> </semantics> <mo> &#10869; </mo> <mn> 1 </mn> </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> &gt; </mo> <mn> 3 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <rem /> <apply> <ci> Binomial </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> $CellContext`p </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> $CellContext`p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> $CellContext`p </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <primes /> </apply> <apply> <gt /> <ci> p </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29