Primality proof for n = 6271:

Take b = 2.

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

19 is prime.
b^((n-1)/19)-1 mod n = 5016, which is a unit, inverse 4702.

11 is prime.
b^((n-1)/11)-1 mod n = 4365, which is a unit, inverse 1701.

(11 * 19) divides n-1.

(11 * 19)^2 > n.

n is prime by Pocklington's theorem.