We prove Cramer's conjecture that $p_{n+1} - p_n = O(\ln(p_n)^2)$, where
$p_n$ is the $n^{th}$ prime and $\ln(x)$ is the natural logarithm of $x$. Also,
Legendre's conjecture follows from this, that is, there exists at least one
prime between two successive square numbers.