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.