Primality proof for n = 12527:

Take b = 2.

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

6263 is prime.
b^((n-1)/6263)-1 mod n = 3, which is a unit, inverse 4176.

(6263) divides n-1.

(6263)^2 > n.

n is prime by Pocklington's theorem.