Proving the irrationality of $e^n$ Let $n$ be a positive integer. I know the traditional proof that $e$ is irrational. How do we show that $e^n$ is irrational in some sort of similar line? I am of course assuming it is but I would be astounded if not. It occurs to me that since $e$ is transcendental, of course $e^n$ is irrational, but I don't want to use that fact.
Googling gives me something for $e^2$, but I could not easily find anything for $e^3$.
 A: I believe there's a proof of this in the book described here.
I  have never personally looked into that, however.
How to show $e^2$ and $e^4$ are irrational is show here.
A: In fact, $e^n$ is transcendental for all $n \in \mathbb{Z}-\{0\}$. To see this it is sufficient to show that $e^n$ is transcendental for all $n \in \mathbb{Z}^+$. If $e^n$ were to be algebraic, this means that there exists a polynomial $p(x)$ with integer coefficients, such that $p(e^n) = 0$. This means that the polynomial $g(x) = p(x^n)$ has $e$ as its root, contradicting the transcendence of $e$.
A: You can find a pretty direct proof of this result here, in proposition 7.
Here is the essence of the argument: for $r(x)$ a polynomial of degree $k$ we can express $$\int_0^1r(x)e^{z-xz}dx=\frac{F(z,0)e^z-F(z,1)}{z^{k+1}}$$
where $F(z,x)$ is a polynomial in $z$ whose coefficients are successive derivatives of $r$ at $x$ (this is proposition 5). In particular, if $e^p=\frac{a}{b}$ is rational and we take $r(x)=x^n(1-x)^n$ for some large $n$ to be specified later, we can use this formula to estimate $aF(p,0)-bF(p,1)\leq ap^{2n+1}$. However, $aF(p,0)-bF(p,1)$ is clearly positive (by looking at the integral), and from the formula for $F(z,x)$ and the fact that many derivatives of $r(x)$ vanish at $0,1$ we can deduce $aF(p,0)-bF(p,1)$ is integer divisible by $n!$. Hence $n!\leq aF(p,0)-bF(p,1)\leq ap^{2n+1}$, so $n!\leq ap^{2n+1}$ which is contradictory for large $n$.
A: Niven's polynomials
Let $f:[0,1]\longrightarrow \mathbb{R}$ , $\displaystyle f(x)=\frac{x^n(1-x)^n}{n!}$ then
$$f(x)=f(1-x)$$  
$$\displaystyle 0\le f(x)<\frac{1}{n!}$$ 
$$f^{(j)}(0)\;,\;f^{(j)}(1) \in \mathbb{Z} \;,\; j\ge 0$$

Proposition. The number $e^3$ is irrational.
Proof: Suppose that $\displaystyle e^3=\frac{a}{b}$ 
$$\displaystyle F=3^{2n}f-3^{2n-1}f'+3^{2n-2}f''-\cdots + f^{(2n)}$$
$$\displaystyle F'+3F=3^{2n+1}f$$ 
$$\displaystyle \mathbb{Z^+} \ni aF(1)-bF(0)=b\Bigl[e^{3x}F(x)\Bigr]_0^1=b\int_0^1 3^{2n+1}e^{3x} f(x)dx \longrightarrow 0^+\;,\;n \longrightarrow\infty  $$ Contradiction
, analogously $e^h$ is irrational for $h \in \mathbb{Z}^+.$
