Primality proof for n = 1277:
Take b = 2.
b^(n-1) mod n = 1.
29 is prime.
b^((n-1)/29)-1 mod n = 963, which is a unit, inverse 61.
11 is prime.
b^((n-1)/11)-1 mod n = 134, which is a unit, inverse 1115.
(11 * 29) divides n-1.
(11 * 29)^2 > n.
n is prime by Pocklington's theorem.