Primality proof for n = 4177:
Take b = 5.
b^(n-1) mod n = 1.
29 is prime.
b^((n-1)/29)-1 mod n = 812, which is a unit, inverse 1785.
3 is prime.
b^((n-1)/3)-1 mod n = 3073, which is a unit, inverse 367.
(3^2 * 29) divides n-1.
(3^2 * 29)^2 > n.
n is prime by Pocklington's theorem.