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• W2908191471 startingPage "1281" @default.
• W2908191471 abstract "It is an open conjecture that for any positive odd integer <italic>m</italic> the function <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C left-parenthesis m right-parenthesis equals left-parenthesis 3 m plus 1 right-parenthesis slash 2 Superscript e left-parenthesis m right-parenthesis Baseline comma> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>3</mml:mn> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>e</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>C(m) = (3m + 1)/{2^{e(m)}},</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e left-parenthesis m right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>e(m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is chosen so that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C left-parenthesis m right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>C(m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is again an odd integer, satisfies <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Superscript h Baseline left-parenthesis m right-parenthesis equals 1> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>h</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>{C^h}(m) = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <italic>h</italic>. Here we show that the number of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m less-than-or-slanted-equals x> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>m leqslant x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which satisfy the conjecture is at least <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x Superscript c> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>c</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{x^c}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a positive constant <italic>c</italic>. A connection between the validity of the conjecture and the diophantine equation <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 Superscript x Baseline minus 3 Superscript y Baseline equals p> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>x</mml:mi> </mml:msup> </mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mn>3</mml:mn> <mml:mi>y</mml:mi> </mml:msup> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>{2^x} - {3^y} = p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is established. It is shown that if the conjecture fails due to an occurrence <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m equals upper C Superscript k Baseline left-parenthesis m right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>m = {C^k}(m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <italic>k</italic> is greater than 17985. Finally, an analogous <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q x plus r> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>qx + r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> problem is settled for certain pairs <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis q comma r right-parenthesis not-equals left-parenthesis 3 comma 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(q,r) ne (3,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W2908191471 created "2019-01-11" @default.
• W2908191471 creator A5028269630 @default.
• W2908191471 date "1978-01-01" @default.
• W2908191471 modified "2023-10-18" @default.
• W2908191471 title "On the “3𝑥+1” problem" @default.
• W2908191471 cites W2013547425 @default.
• W2908191471 cites W2031763298 @default.
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