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http://functions.wolfram.com/04.09.27.0001.02
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ExtendedGCD[m, n] == {GCD[m, n], {r, s}} /; GCD[m, n] == m r + n s &&
Element[Re[m], Integers] && Element[Im[m], Integers] &&
Element[Re[n], Integers] && Element[Im[n], Integers] &&
Element[Re[r], Integers] && Element[Im[r], Integers] &&
Element[Re[s], Integers] && Element[Im[s], Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ExtendedGCD", "[", RowBox[List["m", ",", "n"]], "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]], ",", RowBox[List["{", RowBox[List["r", ",", "s"]], "}"]]]], "}"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["m", " ", "r"]], "+", RowBox[List["n", " ", "s"]]]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", "m", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", "m", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "n", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", "n", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "r", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", "r", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "s", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", "s", "]"]], "\[Element]", "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> <mi> egcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mi> r </mi> <mo> , </mo> <mi> s </mi> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mo> + </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> egcd </ci> <ci> m </ci> <ci> n </ci> </apply> <list> <apply> <gcd /> <ci> m </ci> <ci> n </ci> </apply> <list> <ci> r </ci> <ci> s </ci> </list> </list> </apply> <apply> <and /> <apply> <eq /> <apply> <gcd /> <ci> m </ci> <ci> n </ci> </apply> <apply> <plus /> <apply> <times /> <ci> m </ci> <ci> r </ci> </apply> <apply> <times /> <ci> n </ci> <ci> s </ci> </apply> </apply> </apply> <apply> <in /> <apply> <real /> <ci> m </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <ci> m </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <real /> <ci> n </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <ci> n </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <real /> <ci> r </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <ci> r </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <real /> <ci> s </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <ci> s </ci> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ExtendedGCD", "[", RowBox[List["m_", ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]], ",", RowBox[List["{", RowBox[List["r", ",", "s"]], "}"]]]], "}"]], "/;", RowBox[List[RowBox[List[RowBox[List["GCD", "[", RowBox[List["m", ",", "n"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["m", " ", "r"]], "+", RowBox[List["n", " ", "s"]]]]]], "&&", RowBox[List[RowBox[List["Re", "[", "m", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", "m", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Re", "[", "n", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", "n", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Re", "[", "r", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", "r", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Re", "[", "s", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", "s", "]"]], "\[Element]", "Integers"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
