Primality proof for n = 4977439:
Take b = 2.
b^(n-1) mod n = 1.
2377 is prime. b^((n-1)/2377)-1 mod n = 3063678, which is a unit, inverse 3401230.
(2377) divides n-1.
(2377)^2 > n.
n is prime by Pocklington's theorem.