Primality proof for n = 41389:

Take b = 2.

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

3449 is prime.
b^((n-1)/3449)-1 mod n = 4095, which is a unit, inverse 11785.

(3449) divides n-1.

(3449)^2 > n.

n is prime by Pocklington's theorem.