ExtendedGCD[n, -n] == {Abs[n], {0, -Sign[n]}}

Cell[BoxData[RowBox[List[RowBox[List["ExtendedGCD", "[", RowBox[List["n", ",", RowBox[List["-", "n"]]]], "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], ",", RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", RowBox[List["Sign", "[", "n", "]"]]]]]], "}"]]]], "}"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> egcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> n </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <mo> - </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> egcd </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <list> <apply> <abs /> <ci> n </ci> </apply> <list> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Sign </ci> <ci> n </ci> </apply> </apply> </list> </list> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ExtendedGCD", "[", RowBox[List["n_", ",", RowBox[List["-", "n_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["{", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], ",", RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", RowBox[List["Sign", "[", "n", "]"]]]]]], "}"]]]], "}"]]]]]]

2001-10-29