Primality proof for n = 77946443:
Take b = 2.
b^(n-1) mod n = 1.
8423 is prime.
b^((n-1)/8423)-1 mod n = 687979, which is a unit, inverse 74117547.
661 is prime.
b^((n-1)/661)-1 mod n = 53694906, which is a unit, inverse 17961036.
(661 * 8423) divides n-1.
(661 * 8423)^2 > n.
n is prime by Pocklington's theorem.