Mod[(p - 1)!, p] == p - 1 /; Element[p, Primes]

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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> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </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> <factorial /> <apply> <plus /> <ci> $CellContext`p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> $CellContext`p </ci> </apply> </annotation-xml> </semantics> <mo> &#10869; </mo> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> p </mi> <mo> &#8712; </mo> <semantics> <mi> &#8473; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalP]&quot;, Function[Primes]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <rem /> <apply> <factorial /> <apply> <plus /> <ci> $CellContext`p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> $CellContext`p </ci> </apply> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <in /> <ci> p </ci> <primes /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Mod", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["p_", "-", "1"]], ")"]], "!"]], ",", "p_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["p", "-", "1"]], "/;", RowBox[List["p", "\[Element]", "Primes"]]]]]]]]

2001-10-29