Primality proof for n = 5266351:

Take b = 2.

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

83 is prime.
b^((n-1)/83)-1 mod n = 3188126, which is a unit, inverse 4053108.

47 is prime.
b^((n-1)/47)-1 mod n = 4912638, which is a unit, inverse 852516.

(47 * 83) divides n-1.

(47 * 83)^2 > n.

n is prime by Pocklington's theorem.