Primality proof for n = 6211:
Take b = 2.
b^(n-1) mod n = 1.
23 is prime.
b^((n-1)/23)-1 mod n = 679, which is a unit, inverse 1180.
5 is prime.
b^((n-1)/5)-1 mod n = 3984, which is a unit, inverse 4797.
(5 * 23) divides n-1.
(5 * 23)^2 > n.
n is prime by Pocklington's theorem.