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http://functions.wolfram.com/04.08.02.0002.01
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GCD[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, m]] == p /;
Element[Re[p], Integers] && Element[Im[p], Integers] &&
Element[Re[Subscript[n, k]/p], Integers] && Element[Im[Subscript[n, k]/p],
Integers] && 1 <= k <= m &&
( !Exists[q, Abs[q] > Abs[p] && Element[Re[q], Integers] &&
Element[Im[q], Integers]] && Element[Re[Subscript[n, k]/q],
Integers] && Element[Im[Subscript[n, k]/q], Integers] && 1 <= k <= m)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GCD", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], "]"]], "\[Equal]", "p"]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "p", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", "p", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Re", "[", FractionBox[SubscriptBox["n", "k"], "p"], "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", FractionBox[SubscriptBox["n", "k"], "p"], "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "m"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Not", "[", RowBox[List["Exists", "[", RowBox[List["q", ",", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "q", "]"]], ">", RowBox[List["Abs", "[", "p", "]"]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", "q", "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", "q", "]"]], "\[Element]", "Integers"]]]]]], "]"]], "]"]], "\[And]", RowBox[List[RowBox[List["Re", "[", FractionBox[SubscriptBox["n", "k"], "q"], "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Im", "[", FractionBox[SubscriptBox["n", "k"], "q"], "]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "m"]]]], ")"]]]]]]]]
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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> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mi> p </mi> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> p </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> p </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> <mfrac> <msub> <mi> n </mi> <mi> k </mi> </msub> <mi> p </mi> </mfrac> <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> <mfrac> <msub> <mi> n </mi> <mi> k </mi> </msub> <mi> p </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mi> m </mi> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ¬ </mo> <mrow> <msub> <mo> ∃ </mo> <mi> q </mi> </msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> q </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> > </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> p </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </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> q </mi> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <msub> <mi> n </mi> <mi> k </mi> </msub> <mi> q </mi> </mfrac> <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> <mfrac> <msub> <mi> n </mi> <mi> k </mi> </msub> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <gcd /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <and /> <apply> <in /> <apply> <real /> <ci> p </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <ci> p </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <real /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> k </ci> <ci> m </ci> </apply> <apply> <and /> <apply> <not /> <apply> <exists /> <bvar> <ci> q </ci> </bvar> <apply> <and /> <apply> <gt /> <apply> <abs /> <ci> q </ci> </apply> <apply> <abs /> <ci> p </ci> </apply> </apply> <apply> <in /> <apply> <real /> <ci> q </ci> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <ci> q </ci> </apply> <integers /> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <real /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> k </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GCD", "[", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["n_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["n_", "m_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["p", "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "p", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", "p", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Re", "[", FractionBox[SubscriptBox["n", "k"], "p"], "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", FractionBox[SubscriptBox["n", "k"], "p"], "]"]], "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "m"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["!", RowBox[List[SubscriptBox["\[Exists]", "q"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "q", "]"]], ">", RowBox[List["Abs", "[", "p", "]"]]]], "&&", RowBox[List[RowBox[List["Re", "[", "q", "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", "q", "]"]], "\[Element]", "Integers"]]]], ")"]]]]]], "&&", RowBox[List[RowBox[List["Re", "[", FractionBox[SubscriptBox["n", "k"], "q"], "]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Im", "[", FractionBox[SubscriptBox["n", "k"], "q"], "]"]], "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "m"]]]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
