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• W2022632371 abstract "For any integer <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n greater-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>ngeq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=j left-parenthesis n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>j(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the Jacobsthal function, and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=omega left-parenthesis n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>omega (n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the number of distinct prime divisors of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In 1962 Jacobsthal conjectured that for any integer <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r greater-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>rgeq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the maximal value of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=j left-parenthesis n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>j(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> varies over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper N> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathbb N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=omega left-parenthesis n right-parenthesis equals r> <mml:semantics> <mml:mrow> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>omega (n)=r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is attained when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the product of the first <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> primes. We show that this is true for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r less-than-or-equal-to 23> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>23</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>rleq 23</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and fails at <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r equals 24> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>24</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>r=24</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, thus disproving Jacobsthal’s conjecture." @default.
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• W2022632371 title "Disproof of a conjecture of Jacobsthal" @default.
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