This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An...This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.展开更多
In this paper, we use two new effective tools and ingenious methods to prove the 3X + 1 conjecture. By using the recursive method, we firstly prove that any positive integer can be turned into an element of fourth col...In this paper, we use two new effective tools and ingenious methods to prove the 3X + 1 conjecture. By using the recursive method, we firstly prove that any positive integer can be turned into an element of fourth column of the infinite-row-six-column-matrix after a finite times operation, thus we convert “the 3X + 1 conjecture” into an equivalent conjecture, which is: Any positive integer n must become 1 after finite operations under formation of <span style="white-space:nowrap;">σ(<em>n</em>)</span> , where <img src="Edit_dad9267d-3c54-455b-b30e-63819c207e54.png" width="300" height="117" alt="" /> Then, with the help of the infinite-row-four-column-matrix, we continue to use the recursive method to prove this conjecture strictly.展开更多
The 3<em>X</em> + 1 problem (Collatz conjecture) has been proposed for many years, however no major breakthrough has been made so far. As we know, the Crandall conjecture is a well-known generalization of ...The 3<em>X</em> + 1 problem (Collatz conjecture) has been proposed for many years, however no major breakthrough has been made so far. As we know, the Crandall conjecture is a well-known generalization of the 3<em>X</em> + 1 problem. It is worth noting that, both conjectures are infamous for their simplicity in stating but intractability in solving. In this paper, I aim to provide a clear explanation about the reason why these two problems are difficult to handle and have very different characteristics on convergence of the series via creatively applying the probability theory and global expectancy value <em>E</em>(<em>n</em>) of energy contraction index. The corresponding convergence analysis explicitly shows that <em>a</em> = 3 leads to a difficult problem, while <em>a</em> > 3 leads to a divergent series. To the best of my knowledge, this is the first work to point out the difference between these cases. The corresponding results not only propose a new angle to analyze the 3<em>X</em> + 1 problem, but also shed some light on the future research.展开更多
提出了3x+1的又一推广函数F(z),指出其能引出复杂的分形结构.分析了函数F(z)的基本数学特征,探讨了该映射在C平面上广义M集的图像特征,并绘制了其广义M集的部分美妙的分形图像.利用调色板技术和轨迹井技术结合的方法,绘制F(z)广义M集的...提出了3x+1的又一推广函数F(z),指出其能引出复杂的分形结构.分析了函数F(z)的基本数学特征,探讨了该映射在C平面上广义M集的图像特征,并绘制了其广义M集的部分美妙的分形图像.利用调色板技术和轨迹井技术结合的方法,绘制F(z)广义M集的艺术分形图像,同时在Carlson(Carlson Paul W.Two artistic orbit trap rending methods for Newton M-set fractals[J].Computers & Graphics,1999,23(6):925-931)的基础上提出了环状M集轨迹井,取得了良好的艺术效果,给人一种美的享受.展开更多
文摘This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.
文摘In this paper, we use two new effective tools and ingenious methods to prove the 3X + 1 conjecture. By using the recursive method, we firstly prove that any positive integer can be turned into an element of fourth column of the infinite-row-six-column-matrix after a finite times operation, thus we convert “the 3X + 1 conjecture” into an equivalent conjecture, which is: Any positive integer n must become 1 after finite operations under formation of <span style="white-space:nowrap;">σ(<em>n</em>)</span> , where <img src="Edit_dad9267d-3c54-455b-b30e-63819c207e54.png" width="300" height="117" alt="" /> Then, with the help of the infinite-row-four-column-matrix, we continue to use the recursive method to prove this conjecture strictly.
文摘The 3<em>X</em> + 1 problem (Collatz conjecture) has been proposed for many years, however no major breakthrough has been made so far. As we know, the Crandall conjecture is a well-known generalization of the 3<em>X</em> + 1 problem. It is worth noting that, both conjectures are infamous for their simplicity in stating but intractability in solving. In this paper, I aim to provide a clear explanation about the reason why these two problems are difficult to handle and have very different characteristics on convergence of the series via creatively applying the probability theory and global expectancy value <em>E</em>(<em>n</em>) of energy contraction index. The corresponding convergence analysis explicitly shows that <em>a</em> = 3 leads to a difficult problem, while <em>a</em> > 3 leads to a divergent series. To the best of my knowledge, this is the first work to point out the difference between these cases. The corresponding results not only propose a new angle to analyze the 3<em>X</em> + 1 problem, but also shed some light on the future research.
文摘提出了3x+1的又一推广函数F(z),指出其能引出复杂的分形结构.分析了函数F(z)的基本数学特征,探讨了该映射在C平面上广义M集的图像特征,并绘制了其广义M集的部分美妙的分形图像.利用调色板技术和轨迹井技术结合的方法,绘制F(z)广义M集的艺术分形图像,同时在Carlson(Carlson Paul W.Two artistic orbit trap rending methods for Newton M-set fractals[J].Computers & Graphics,1999,23(6):925-931)的基础上提出了环状M集轨迹井,取得了良好的艺术效果,给人一种美的享受.
基金Supported by the National Natural Science Foundation of China(11371207)the Natural Science Foundation of Jiangsu Province(BK20140424)the Natural Science Foundation of Nantong University(14ZY009)
