Primality proof for n = 9151:
Take b = 2.
b^(n-1) mod n = 1.
61 is prime.
b^((n-1)/61)-1 mod n = 1524, which is a unit, inverse 3921.
5 is prime.
b^((n-1)/5)-1 mod n = 2004, which is a unit, inverse 516.
(5^2 * 61) divides n-1.
(5^2 * 61)^2 > n.
n is prime by Pocklington's theorem.