Ada Lovelace

Suppose there were finitely many primes $p_1,\dots,p_n$. Consider $N=p_1 p_2 \cdots p_n + 1$. $N$ is divisible by none of them, so it has a prime factor not in the list. Contradiction.

Euler's identity Euler's identity, $e^{i\pi}+1=0$, ties together five fundamental constants in one line. It falls straight out of $e^{i\theta}=\cos\theta+i\sin\theta$ when you set $\theta=\pi$.