The 3x + 1 problem, is a math problem that has baffled mathematicians for over 50 years. It’s easy to explain: take any positive number, if it’s even, divide it by 2;if it’s odd, multiply it by 3 and add 1. Repeat ...The 3x + 1 problem, is a math problem that has baffled mathematicians for over 50 years. It’s easy to explain: take any positive number, if it’s even, divide it by 2;if it’s odd, multiply it by 3 and add 1. Repeat this process with the resulting number, and the conjecture says that you will eventually reach 1. Despite testing all starting values up to an enormous number, no one has proved the conjecture is true for all possible starting values. The problem’s importance lies in its simplicity and difficulty, inspiring new ideas in mathematics and advancing fields like number theory, dynamical systems, and computer science. Proving or disproving the conjecture would revolutionize our understanding of math. The presence of infinite sequences is a matter of question. To investigate and solve this conjecture, we are utilizing a novel approach involving the fields of number theory and computer science.展开更多
This paper studies the property of the recursive sequences in the 3x ＋ 1 conjecture. The authors introduce the concept of μ function, with which the 3x ＋ 1 conjecture can be transformed into two other conjectures：...This paper studies the property of the recursive sequences in the 3x ＋ 1 conjecture. The authors introduce the concept of μ function, with which the 3x ＋ 1 conjecture can be transformed into two other conjectures： one is eventually periodic conjecture of the μ function and the other is periodic point conjecture. The authors prove that the 3x ＋ 1 conjecture is equivalent to the two conjectures above. In 2007, J. L. Simons proved the non-existence of nontrivial 2-cycle for the T function. In this paper, the authors prove that the μ function has nol-periodic points for 2 ≤ 1 ≤12. In 2005, J. L. Simons and B. M. M de Weger proved that there is no nontrivial/-cycle for the T function for 1 ≤68, and in this paper, the authors prove that there is no nontrivial l-cycle for the μ function for 2 ≤ 1≤ 102.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
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
文摘The 3x + 1 problem, is a math problem that has baffled mathematicians for over 50 years. It’s easy to explain: take any positive number, if it’s even, divide it by 2;if it’s odd, multiply it by 3 and add 1. Repeat this process with the resulting number, and the conjecture says that you will eventually reach 1. Despite testing all starting values up to an enormous number, no one has proved the conjecture is true for all possible starting values. The problem’s importance lies in its simplicity and difficulty, inspiring new ideas in mathematics and advancing fields like number theory, dynamical systems, and computer science. Proving or disproving the conjecture would revolutionize our understanding of math. The presence of infinite sequences is a matter of question. To investigate and solve this conjecture, we are utilizing a novel approach involving the fields of number theory and computer science.
基金supported by Natural Science Foundation of China under Grant Nos.60833008 and 60902024
文摘This paper studies the property of the recursive sequences in the 3x ＋ 1 conjecture. The authors introduce the concept of μ function, with which the 3x ＋ 1 conjecture can be transformed into two other conjectures： one is eventually periodic conjecture of the μ function and the other is periodic point conjecture. The authors prove that the 3x ＋ 1 conjecture is equivalent to the two conjectures above. In 2007, J. L. Simons proved the non-existence of nontrivial 2-cycle for the T function. In this paper, the authors prove that the μ function has nol-periodic points for 2 ≤ 1 ≤12. In 2005, J. L. Simons and B. M. M de Weger proved that there is no nontrivial/-cycle for the T function for 1 ≤68, and in this paper, the authors prove that there is no nontrivial l-cycle for the μ function for 2 ≤ 1≤ 102.
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.
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.
