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.