39,916,801 has just been discovered to be the largest prime number.
$11$ is a positive integer such that $11! + 1$ is prime; let $n = 11!$.
Now, $n+2$ is clearly a multiple of 2, $n+3$ is a multiple of 3 and so on up to $2n$. Applying a similar argument accounts for all the numbers between $2n$ and $3n$, (e.g., $2n + 5$ is clearly a multiple of 5), between $3n$ and $4n$ and - in general - between $an$ and $(a+1)n$.
Thus, $11! + 1 = 39,916,801$ is the largest prime number.