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• W2092160372 abstract "For relatively prime positive integers <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u 0> <mml:semantics> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>u_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding=application/x-tex>r</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we consider the arithmetic progression <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace u Subscript k Baseline colon equals u 0 plus k r right-brace Subscript k equals 0 Superscript n> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>:=</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>k</mml:mi> <mml:mi>r</mml:mi> <mml:msubsup> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:msubsup> </mml:mrow> <mml:annotation encoding=application/x-tex>{u_k:=u_0+kr}_{k=0}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Define <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Subscript n Baseline colon equals l c m left-brace right-brace comma comma period period period comma u 0 comma u 1 comma period period period comma un> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>:=</mml:mo> <mml:mi>lcm</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L_n:=operatorname {lcm}{u_0, u_1, ..., u_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a greater-than-or-equal-to 2> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>age 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be any integer. In this paper, we show that for integers <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=alpha comma r greater-than-or-equal-to a> <mml:semantics> <mml:mrow> <mml:mi>α<!-- α --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mi>a</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>alpha , rgeq a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n greater-than-or-equal-to 2 alpha r> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> <mml:mi>α<!-- α --></mml:mi> <mml:mi>r</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>ngeq 2alpha r</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we have <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Subscript n Baseline greater-than-or-equal-to u 0 r Superscript alpha plus a minus 2 Baseline left-parenthesis r plus 1 right-parenthesis Superscript n Baseline period> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>α<!-- α --></mml:mi> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L_ngeq u_0r^{alpha +a-2}(r+1)^n.</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> In particular, letting <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a equals 2> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>a=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> yields an improvement to the best previous lower bound on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Subscript n> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>L_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (obtained by Hong and Yang) for all but three choices of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=alpha comma r greater-than-or-equal-to 2> <mml:semantics> <mml:mrow> <mml:mi>α<!-- α --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>alpha , rgeq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W2092160372 title "Further improvements of lower bounds for the least common multiples of arithmetic progressions" @default.
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