Primality proof for n = 2377:
Take b = 2.
b^(n-1) mod n = 1.
11 is prime.
b^((n-1)/11)-1 mod n = 1719, which is a unit, inverse 1044.
3 is prime.
b^((n-1)/3)-1 mod n = 1654, which is a unit, inverse 240.
(3^3 * 11) divides n-1.
(3^3 * 11)^2 > n.
n is prime by Pocklington's theorem.