Primality proof for n = 41413:

Take b = 2.

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

29 is prime.
b^((n-1)/29)-1 mod n = 6554, which is a unit, inverse 32611.

17 is prime.
b^((n-1)/17)-1 mod n = 33294, which is a unit, inverse 26478.

(17 * 29) divides n-1.

(17 * 29)^2 > n.

n is prime by Pocklington's theorem.