Primality proof for n = 29311:

Take b = 2.

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

977 is prime.
b^((n-1)/977)-1 mod n = 21271, which is a unit, inverse 10175.

(977) divides n-1.

(977)^2 > n.

n is prime by Pocklington's theorem.