Primality proof for n = 2543:
Take b = 2.
b^(n-1) mod n = 1.
41 is prime.
b^((n-1)/41)-1 mod n = 816, which is a unit, inverse 1499.
31 is prime.
b^((n-1)/31)-1 mod n = 751, which is a unit, inverse 2201.
(31 * 41) divides n-1.
(31 * 41)^2 > n.
n is prime by Pocklington's theorem.