Primality proof for n = 1123:
Take b = 2.
b^(n-1) mod n = 1.
17 is prime.
b^((n-1)/17)-1 mod n = 612, which is a unit, inverse 934.
11 is prime.
b^((n-1)/11)-1 mod n = 847, which is a unit, inverse 1005.
(11 * 17) divides n-1.
(11 * 17)^2 > n.
n is prime by Pocklington's theorem.