Primality proof for n = 68477:

Take b = 2.

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

53 is prime.
b^((n-1)/53)-1 mod n = 67193, which is a unit, inverse 38025.

19 is prime.
b^((n-1)/19)-1 mod n = 60449, which is a unit, inverse 62890.

(19 * 53) divides n-1.

(19 * 53)^2 > n.

n is prime by Pocklington's theorem.