Primality proof for n = 6997:

Take b = 2.

b^(n-1) mod n = 1.

53 is prime.
b^((n-1)/53)-1 mod n = 1000, which is a unit, inverse 4667.

11 is prime.
b^((n-1)/11)-1 mod n = 5165, which is a unit, inverse 3044.

(11 * 53) divides n-1.

(11 * 53)^2 > n.

n is prime by Pocklington's theorem.