Primality proof for n = 11171:

Take b = 2.

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

1117 is prime.
b^((n-1)/1117)-1 mod n = 1023, which is a unit, inverse 2861.

(1117) divides n-1.

(1117)^2 > n.

n is prime by Pocklington's theorem.