Primality proof for n = 41:

Take b = 3.

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

2 is prime.
b^((n-1)/2)-1 mod n = 39, which is a unit, inverse 20.

(2^3) divides n-1.

(2^3)^2 > n.

n is prime by Pocklington's theorem.