$e$无理数的几何证明及其无理数的一个新度量
A geometric proof that $e$ is irrational and a new measure of its irrationality
摘要：
我们给出了一个简单的几何证明，$e$是无理的，使用一个结构的嵌套序列的闭区间的交集$e$。这个证明导致了$e$的一个新的非理性度量：如果$p$和$q$是整数，且$q > 1$，那么$|e - p/q| > 1/(S(q)+1)!$，其中$S(q)$是最小的正整数，使得$S(q)!$是$q$的倍数。我们将此措施$e$一个已知的和最大的素数因子的整数。我们做了两个假设，并回顾了康托的一个定理，可以证明一个类似的结构。
英文摘要：
We give a simple geometric proof that $e$ is irrational, using a construction of a nested sequence of closed intervals with intersection $e$. The proof leads to a new measure of irrationality for $e$: if $p$ and $q$ are integers with $q > 1$, then $|e - p/q| > 1/(S(q)+1)!$, where $S(q)$ is the smallest positive integer such that $S(q)!$ is a multiple of $q$. We relate this measure for $e$ to a known one and to the greatest prime factor of an integer. We make two conjectures and recall a theorem of Cantor that can be proved by a similar construction.
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