Primality proof for n = 521:

Take b = 2.

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

13 is prime.
b^((n-1)/13)-1 mod n = 254, which is a unit, inverse 80.

5 is prime.
b^((n-1)/5)-1 mod n = 515, which is a unit, inverse 434.

(5 * 13) divides n-1.

(5 * 13)^2 > n.

n is prime by Pocklington's theorem.