Primality proof for n = 2663886769:

Take b = 2.

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

227 is prime.
b^((n-1)/227)-1 mod n = 676872375, which is a unit, inverse 2648350256.

89 is prime.
b^((n-1)/89)-1 mod n = 1442155822, which is a unit, inverse 2563536608.

67 is prime.
b^((n-1)/67)-1 mod n = 189440380, which is a unit, inverse 2419164473.

(67 * 89 * 227) divides n-1.

(67 * 89 * 227)^2 > n.

n is prime by Pocklington's theorem.