Primality proof for n = 5413:
Take b = 2.
b^(n-1) mod n = 1.
41 is prime.
b^((n-1)/41)-1 mod n = 973, which is a unit, inverse 484.
11 is prime.
b^((n-1)/11)-1 mod n = 4477, which is a unit, inverse 4933.
(11 * 41) divides n-1.
(11 * 41)^2 > n.
n is prime by Pocklington's theorem.