Primality proof for n = 1163251:
Take b = 2.
b^(n-1) mod n = 1.
47 is prime.
b^((n-1)/47)-1 mod n = 566211, which is a unit, inverse 963760.
11 is prime.
b^((n-1)/11)-1 mod n = 911772, which is a unit, inverse 360291.
3 is prime.
b^((n-1)/3)-1 mod n = 89984, which is a unit, inverse 745505.
(3^2 * 11 * 47) divides n-1.
(3^2 * 11 * 47)^2 > n.
n is prime by Pocklington's theorem.