Primality proof for n = 2531:
Take b = 2.
b^(n-1) mod n = 1.
23 is prime.
b^((n-1)/23)-1 mod n = 216, which is a unit, inverse 996.
11 is prime.
b^((n-1)/11)-1 mod n = 1054, which is a unit, inverse 1083.
(11 * 23) divides n-1.
(11 * 23)^2 > n.
n is prime by Pocklington's theorem.