Primality proof for n = 23311:

Take b = 3.

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

37 is prime.
b^((n-1)/37)-1 mod n = 11310, which is a unit, inverse 14776.

7 is prime.
b^((n-1)/7)-1 mod n = 7031, which is a unit, inverse 3710.

(7 * 37) divides n-1.

(7 * 37)^2 > n.

n is prime by Pocklington's theorem.