Primality proof for n = 1171:
Take b = 2.
b^(n-1) mod n = 1.
13 is prime.
b^((n-1)/13)-1 mod n = 165, which is a unit, inverse 951.
5 is prime.
b^((n-1)/5)-1 mod n = 69, which is a unit, inverse 594.
(5 * 13) divides n-1.
(5 * 13)^2 > n.
n is prime by Pocklington's theorem.