e^(pi/3) in base 16; average of the iterated Gauss map of a half-regular continued fraction; 1000th digit of e^(pi/3) is e^(pi/3) irrational?

可以利用导数证明当 x\in(0,e) 时 \frac{lnx}{x}<\frac{1}{e} 令 x=\frac{e^2}{π} ，则 ln\frac{e^2}{π}<\frac{e}{π} 故 2-lnπ<\frac{e}{π} 进而可以得到 elnπ>2e-\frac{e^2}{π}>3 . ∴ ln(π^e)>ln(e^3) 进而我们完成了证明

In mathematics, the exponential of pi eπ, [1] also called Gelfond's constant,[2] is the real number e raised to the power π. Its decimal expansion is given by: Like both e and π, this constant is both irrational and transcendental.

2018年3月30日 · The problematic step is when you say "the cubic root of $-1$ is $-1$ itself". Yes, $(-1)^3 = -1$, but there are other cube roots of $-1$, namely $e^{i(\pi/3)}$, for example. The problem is that $(-1)^{1/3}$ is multi-valued.

\int e^x\cos (x)dx \int_{0}^{\pi}\sin(x)dx \sum_{n=0}^{\infty}\frac{3}{2^n} Show More

2022年9月23日 · 拿出一张草稿纸，列竖式，先计算 (e^3)^3\leq (2.72)^9 \approx8149 你只要知道 e\approx 2.718 即可. 再拿出一张草稿纸（或者如果你原来用的是A3的草稿纸，把它反过来），列竖式，计算 (\pi ^e)^3 \geq \pi ^8 \geq 3.1^8 \approx 8529. 放大后的 （e^3)^3 小于缩小后的 (\pi^e)^3 ，故后者 ...

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To finish, we just need to show the initial inequality. The function f: x ↦ x2 2x − 3 is decreasing for 3 2 <x <3, so we'll need a lower bound for e. From the series e = ∑∞k = 0 1 k! we see e> 8 / 3.

典型证明的套路就是谈笑间 “注意到” 一个关于 e 或者 \pi 的等于一个大于0的定积分，例如通过以下方法证明 e^3>20 ： 注意到. \begin {align*}&e^3-20 = \frac {1} {8} \int_0^1x^3 (1-x)^4 (28+33x)e^ {3x} dx > 0.\end {align*} 证毕！ 下面的评论基本都是“ 注意力惊人 ”、“ 中国人自己的 拉马努金 ”、“ Attention is all you need! ”等。 方法看起来非常的酷炫和优雅，但其实原理并不复杂。 有些结果看起来非常“鬼畜”，但是借助软件来计算也不困难。 本文来介绍具体的构造方法。 …

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