An E-Sequence Approach to the 3<i>x</i> + 1 Problem

• Main Author: Sanmin Wang
• Format: Article
• Language: English
• Published: MDPI AG 2019-11-01
• Series: Symmetry
• Subjects: 3<i>x</i> + 1 problem; e-sequence approach; ω-divergence of non-periodic e-sequences; wendel’s inequality
• Online Access: https://www.mdpi.com/2073-8994/11/11/1415

For any odd positive integer <i>x</i>, define <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>0</mn> </mrow> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>1</mn> </mrow> </msub> </semantics> </math> </inline-formula> by setting <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mo>&nbsp;</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <msub> <mi>x</mi> <mrow> <mi>n</mi> <mo>&#8722;</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> <msup> <mn>2</mn> <msub> <mi>a</mi> <mi>n</mi> </msub> </msup> </mfrac> </mrow> </semantics> </math> </inline-formula> such that all <inline-formula> <math display="inline"> <semantics> <msub> <mi>x</mi> <mi>n</mi> </msub> </semantics> </math> </inline-formula> are odd. The <inline-formula> <math display="inline"> <semantics> <mrow> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> problem asserts that there is an <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> for all <i>x</i>. Usually, <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>0</mn> </mrow> </msub> </semantics> </math> </inline-formula> is called the trajectory of <i>x</i>. In this paper, we concentrate on <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>1</mn> </mrow> </msub> </semantics> </math> </inline-formula> and call it the E-sequence of <i>x</i>. The idea is that we generalize E-sequences to all infinite sequences <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>1</mn> </mrow> </msub> </semantics> </math> </inline-formula> of positive integers and consider all these generalized E-sequences. We then define <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>1</mn> </mrow> </msub> </semantics> </math> </inline-formula> to be <inline-formula> <math display="inline"> <semantics> <mi mathvariant="sans-serif">&#937;</mi> </semantics> </math> </inline-formula>-convergent to <i>x</i> if it is the E-sequence of <i>x</i> and to be <inline-formula> <math display="inline"> <semantics> <mi mathvariant="sans-serif">&#937;</mi> </semantics> </math> </inline-formula>-divergent if it is not the E-sequence of any odd positive integer. We prove a remarkable fact that the <inline-formula> <math display="inline"> <semantics> <mi mathvariant="sans-serif">&#937;</mi> </semantics> </math> </inline-formula>-divergence of all non-periodic E-sequences implies the periodicity of <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>0</mn> </mrow> </msub> </semantics> </math> </inline-formula> for all <inline-formula> <math display="inline"> <semantics> <msub> <mi>x</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula>. The principal results of this paper are to prove the <inline-formula> <math display="inline"> <semantics> <mi mathvariant="sans-serif">&#937;</mi> </semantics> </math> </inline-formula>-divergence of several classes of non-periodic E-sequences. Especially, we prove that all non-periodic E-sequences <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>1</mn> </mrow> </msub> </semantics> </math> </inline-formula> with <inline-formula> <math display="inline"> <semantics> <mrow> <mstyle displaystyle="true"> <munder> <mover> <mo movablelimits="true" form="prefix">lim</mo> <mo>&#175;</mo> </mover> <mrow> <mi>n</mi> <mo>&#8594;</mo> <mo>&#8734;</mo> </mrow> </munder> </mstyle> <mfrac> <msub> <mi>b</mi> <mi>n</mi> </msub> <mi>n</mi> </mfrac> <mo>&gt;</mo> <msub> <mo form="prefix">log</mo> <mn>2</mn> </msub> <mn>3</mn> </mrow> </semantics> </math> </inline-formula> are <inline-formula> <math display="inline"> <semantics> <mi mathvariant="sans-serif">&#937;</mi> </semantics> </math> </inline-formula>-divergent by using Wendel&#8217;s inequality and the Matthews and Watts&#8217; formula <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msup> <mn>3</mn> <mi>n</mi> </msup> <msub> <mi>x</mi> <mn>0</mn> </msub> </mrow> <msup> <mn>2</mn> <msub> <mi>b</mi> <mi>n</mi> </msub> </msup> </mfrac> <mstyle displaystyle="true"> <munderover> <mo>&#8719;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>n</mi> <mo>&#8722;</mo> <mn>1</mn> </mrow> </munderover> </mstyle> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <msub> <mi>x</mi> <mi>k</mi> </msub> </mrow> </mfrac> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>b</mi> <mi>n</mi> </msub> <mo>=</mo> <mstyle displaystyle="true"> <munderover> <mo>&#8721;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> </mstyle> <msub> <mi>a</mi> <mi>k</mi> </msub> </mrow> </semantics> </math> </inline-formula>. These results present a possible way to prove the periodicity of trajectories of all positive integers in the <inline-formula> <math display="inline"> <semantics> <mrow> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> problem, and we call it the E-sequence approach.