3x+ 1 猜想的等价集

本文定义了３ｘ＋１猜想的等价集,并给出３ｘ＋１猜想的一个等价命题,构造出渐近密度可以任意小的３ｘ＋１猜想等价集。     

关于“3x＋1”问题的研究

本文分析了“３ｘ＋１”问题中存在的结构，进而指出了除ｘ＝１之外，“３ｘ＋１”问题不存在循环。     

关于3x＋1问题的推广

推广了3x 1问题，得到一个一般意义的运算法则，用计算机验证了这个猜测。     

关于3x+1问题的一点注记

给出了3x 1问题的三个等价命题,其中构造数列{T^(k)(n)2∑^k-1i=0i^x(n)}k=1 ∞,证明它是一个单调递减下有界数列。     

3N+1猜想与3N+3k猜想的等价性及相关性质

将数论中3N＋1猜想推广为3N＋3^k猜想．得到了3N＋1猜想与3N＋3^k猜想的等价性．得到有关3N＋3猜想的一些性质．3N＋1猜想的推广、3N＋3猜想的一些性质的建立对于研究4K＋3型奇数在3N＋3猜想压缩迭代中起到简化作用,同时也为3N＋1猜想的研究提供了新思路．     

不可能只用Peano公理组证明3x+1猜想

在本文中,我们用模型论方法证明了:存在一个Peano公理组的模型,在其中3x+1猜想不成立.从而可知:不可能只用Peano公理组证明3x+1猜想.     

3 N+1猜想中的伸长迭代

提出了伸长迭代的概念;给出了该迭代下的某些结果,其中包括：a.关于数集与奇偶矢量集的对应问题;b．关于l-tuple的不变性;c．n的项公式的证明;d．关于3N＋1猜想的等价命题;e．关于系数停止次数tc(n）的性质等。     

3N+1猜想中tc(n)与ta(n)的相等性

讨论了3N 1猜想中n的系数停止次数tc(n)和足够停止次数tc(n)的相等性。证明了当d=∑i=1^k=1xi(n)不是很大时tc(n)和ta(n)是相等的。由此有理由猜想，当d不满足界定条件时，也有tc(n)=ta(n)。     

对叙拉古猜想的一些讨论

本文对使叙拉古猜想不成立的可凝数给出了较「１」更精确的结论,还导出一个递推关系,并用三进数制简化了叙拉古运算。     

Dynamics of the generalized 3x + 1 function determined by its fractal images

Two different complex maps were obtained by generalizing 3x+1 function to the complex plane, and fractal images for this two complex maps were constructed by using escape time, stopping time and total stopping time arithmetic. The dynamics of the generalized 3x+1 function based on the structural characteristics of the fractal images was studied. We found that: (1) The size and structure of the stable regions, stopping regions, total stopping regions, and divergent regions for the three types of fractal images depend on convergence rate of the map on the x and y axes. (2) The black stable regions constructed respectively by escape time and total stopping time are almost overlapped, demonstrating that 3x + 1 function converged steadily. (3) All of the three fractal images are symmetric to the real axis. The structures on the neighborhood of positive integer number are symmetric to a perpendicular line, which is corresponding to the point or its nearby points on the x axis. And the structures have complicated fractal structure characteristics. These findings indicate that the generalized 3x + 1 function on integer number and its neighborhood contains plentiful information in the complex plane.     

Golomb猜想在有限域GF（p^n）中的若干结果

本文用比较简捷的方法获得了Golomb猜想在有限域GF(p~n)中成立的几个结果。这些结果对于不太大的有限域GF(p~n)来说是有意义的,对这些有限域的代数构造是有价值的。     

3N+1猜想中周期数的研究

给出了关于3N 1猜想中周期数存在的一个必要条件：S^l(mxi)=nxi,并在此基础上推广为S^l(σbxi=n^b 1xi,且给出周期数x1的具体表达式x1=r1／(1-3^l／2^k)。证明了圈长为2和3的周期数不存在。     

关于不定方程y（y+1）（y+2）（y+3）=4p2kx（x+1）（x+2）（x+3）

用初等方法证明了不定方程y(y+1)(y+2)(y+3)=nx(x+1)(x+2)(x+3)在n=4p2k(p为奇素数,k为正整数)时无正整数解(x,y).     

Dynamics of the generalized 3x + 1 function determined by its fractal images

Two different complex maps were obtained by generalizing 3x ＋ 1 function to the complex plane, and fractal images for these two complex maps were constructed by using escape time, stopping time and total stopping time arithmetic. The dynamics of the generalized 3x＋1 function based on the structural characteristics of the fractal images was studied. We found that：（1） the size and structure of the stable regions, stopping regions, total stopping regions, and divergent regions for the three types of fractal images depend on convergence rate of the map on the x and y axes. （2） The black stable regions constructed, respectively, by escape time and total stopping time are almost overlapped, demonstrating that 3x ＋1 function converged steadily. （3） All of the three fractal images are symmetric to the real axis. The structures on the neighborhood of positive integer number are symmetric to a perpendicular line, which is corresponding to the point or its nearby points on the x axis. And the structures have complicated fractal structure characteristics. These findings indicate that the generalized 3x＋1 function on integer number and its neighborhood contains plentiful information in the complex plane.