Primality proof for n = 12611:
Take b = 2.
b^(n-1) mod n = 1.
97 is prime.
b^((n-1)/97)-1 mod n = 10523, which is a unit, inverse 7749.
13 is prime.
b^((n-1)/13)-1 mod n = 5484, which is a unit, inverse 7453.
(13 * 97) divides n-1.
(13 * 97)^2 > n.
n is prime by Pocklington's theorem.