Primality proof for n = 577547:

Take b = 2.

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

288773 is prime.
b^((n-1)/288773)-1 mod n = 3, which is a unit, inverse 192516.

(288773) divides n-1.

(288773)^2 > n.

n is prime by Pocklington's theorem.