# Proving the primality of these large numbers?

In 2007, Vautier claimed that the largest known consecutive pair of prime numbers (at the time) was $2003663613\cdot2^{195000}-1$ and $2003663613\cdot2^{195000}+1$.

I was wondering how Vautier found these two extremely large numbers to be prime numbers and if there was any way I could determine if a random and extremely large number were prime.

Proth's theorem allows very fast proofs for numbers of the form $k \times 2^n+1$ ($k$ odd, $k < 2^n$).
Lucas-Lehmer-Riesel allows fast proof for numbers of the form $k \times 2^n-1$ ($k < 2^n$).