Primality proof for n = 3134327699:

Take b = 2.

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

5479 is prime.
b^((n-1)/5479)-1 mod n = 1470288163, which is a unit, inverse 1329895913.

2777 is prime.
b^((n-1)/2777)-1 mod n = 13143832, which is a unit, inverse 863989761.

(2777 * 5479) divides n-1.

(2777 * 5479)^2 > n.

n is prime by Pocklington's theorem.