论文标题
一系列代数整数关系编号，收敛到4
A sequence of algebraic integer relation numbers which converges to 4
论文作者
论文摘要
令$α\在\ Mathbb {r} $中，让$$ a = \ begin {bmatrix} 1＆1＆1＆1＆1＆1 \ end {bmatrix} \ \ \ \ \ \ \ \ text {and} \b_α= \ \ a = \ \ bmatrix {bmatrix} 1＆0＆0＆0＆0 \ \ \ \ $ 1 \ $ unecriack $g_α$ of $ \ mathrm {sl} _2（\ mathbb {r}）$是由矩阵$ a $和$b_α$生成的组。在本文中，我们调查了组$g_α的属性。$我们构建了子组$g_α的Farey图的概括。$此图确定了组$g_α$是免费等级$ 2 $的组。更确切地说，$g_α$是一组自由等级$ 2 $的组，并且仅当图为树时。特别是，我们表明，如果$ 1/2 $是图的顶点，那么$g_α$不是排名$ 2 $的免费组。使用此功能，我们构建了一个实数序列，以使序列收敛到$ 4 $，并且每个数字都有相应的组，该组不是排名$ 2 $的免费组。事实证明，实数是代数整数。
Let $α\in \mathbb{R}$ and let $$A=\begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix} \ \text{and} \ B_α = \begin{bmatrix} 1 & 0 \\ α& 1\end{bmatrix}.$$ The subgroup $G_α$ of $\mathrm{SL}_2(\mathbb{R})$ is a group generated by the matrices $A$ and $B_α$. In this paper, we investigate the property of the group $G_α.$ We construct a generalization of the Farey graph for the subgroup $G_α.$ This graph determines whether the group $G_α$ is a free group of rank $2$. More precisely, the group $G_α$ is a free group of rank $2$ if and only if the graph is tree. In particular, we show that if $1/2$ is a vertex of the graph, then $G_α$ is not a free group of rank $2$. Using this, we construct a sequence of real numbers so that the sequence converges to $4$ and each number has the corresponding group that is not a free group of rank $2$. It turns out that the real numbers are algebraic integers.
