Primality proof for n = 4013:
Take b = 2.
b^(n-1) mod n = 1.
59 is prime.
b^((n-1)/59)-1 mod n = 3465, which is a unit, inverse 1179.
17 is prime.
b^((n-1)/17)-1 mod n = 4002, which is a unit, inverse 1824.
(17 * 59) divides n-1.
(17 * 59)^2 > n.
n is prime by Pocklington's theorem.