Largest 9-Digit Prime Number

Theorem
The largest prime number with $9$ digits is $999 \, 999 \, 937$.

Proof
Consider the numbers $\sqbrk {999 \, 999 \, 9ab}$.

Since $999 \, 999 \, 000$ is divisible by $2, 3, 5, 7, 11, 13$,

if $\sqbrk {9ab}$ is divisible by these primes, so is $\sqbrk {999 \, 999 \, 9ab}$.

After this elimination the only $\sqbrk {ab} > 37$ that remains are:


 * $41, 43, 47, 53, 61, 67, 71, 77, 83, 89, 91, 97$

We have:

And we do have $999 \, 999 \, 937$ is prime.