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.