ExtendedGCD[m + k n, n][[2,1]] == ExtendedGCD[m, n][[2,1]] /; Element[k, Integers]

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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> <mrow> <mrow> <mi> egcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> + </mo> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mrow> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#12314; </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> &#12315; </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> egcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#12314; </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> &#12315; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <selector /> <apply> <ci> egcd </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <ci> k </ci> <ci> n </ci> </apply> </apply> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> <cn type='integer'> 1 </cn> </apply> <apply> <selector /> <apply> <ci> egcd </ci> <ci> m </ci> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <in /> <ci> k </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ExtendedGCD", "[", RowBox[List[RowBox[List["m_", "+", RowBox[List["k_", " ", "n_"]]]], ",", "n_"]], "]"]], "\[LeftDoubleBracket]", RowBox[List["2", ",", "1"]], "\[RightDoubleBracket]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["ExtendedGCD", "[", RowBox[List["m", ",", "n"]], "]"]], "\[LeftDoubleBracket]", RowBox[List["2", ",", "1"]], "\[RightDoubleBracket]"]], "/;", RowBox[List["k", "\[Element]", "Integers"]]]]]]]]

2001-10-29