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.