论文标题
分数$ 3N+1 $猜想
A Fractional $3n+1$ Conjecture
论文作者
论文摘要
在本文中，我们介绍并讨论\ emph {量化数}的顺序定义为$ u_0 \ in \ \ m m mathbb r $和$ u_ {n+1} =δ（u_n）$ where \ begin \ begin {quation*}δ（equation*}δ（x） \ operatorName {frac}（x）<\ frac {1} {2} \\ [4px] \ frac {3x+1} {2} {2}＆\ text {if} \ operatorNAMe {frac}（frac}（frac}（frac}（x）\ geq \ geq \ frac {1}让人想起著名的柯拉茨序列，似乎表现出有趣的行为。的确，我们猜测迭代$δ$最终将收敛至零，或者循环整数零件$ 1,2,4,7,11,18,9,4,4,4,7,3,5,5,9,4,4,1118,118,9,9,9,3,3,3,3,3,3,3,3,2,22 o os of。 我们证明了[0，100] $中$ u_0 \的猜想。将证据扩展到较大的固定值似乎是计算能力的问题。作者承诺向完全证明或反驳猜想的第一个人提供奖励 - 并在认真的指导数学会议或期刊上发表了证明。
In this paper we introduce and discuss the sequence of \emph{real numbers} defined as $u_0 \in \mathbb R$ and $u_{n+1} = Δ(u_n)$ where \begin{equation*} Δ(x) = \begin{cases} \frac{x}{2} &\text{if } \operatorname{frac}(x)<\frac{1}{2} \\[4px] \frac{3x+1}{2} & \text{if } \operatorname{frac}(x)\geq\frac{1}{2} \end{cases} \end{equation*} This sequence is reminiscent of the famous Collatz sequence, and seems to exhibit an interesting behaviour. Indeed, we conjecture that iterating $Δ$ will eventually either converge to zero, or loop over sequences of real numbers with integer parts $1,2,4,7,11,18,9,4,7,3,5,9,4,7,11,18,9,4,7,3,6,3,1,2,4,7,3,6,3$. We prove this conjecture for $u_0 \in [0, 100]$. Extending the proof to larger fixed values seems to be a matter of computing power. The authors pledge to offer a reward to the first person who proves or refutes the conjecture completely -- with a proof published in a serious refereed mathematical conference or journal.
