Primality proof for n = 2746144996771313789:

Take b = 2.

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

15643211 is prime.
b^((n-1)/15643211)-1 mod n = 1155916230020495012, which is a unit, inverse 555212053675919679.

34483 is prime.
b^((n-1)/34483)-1 mod n = 1595067458926635507, which is a unit, inverse 460538071750631915.

(34483 * 15643211) divides n-1.

(34483 * 15643211)^2 > n.

n is prime by Pocklington's theorem.