Primality proof for n = 5089297693:

Take b = 2.

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

4523 is prime.
b^((n-1)/4523)-1 mod n = 2688513124, which is a unit, inverse 1328467536.

2287 is prime.
b^((n-1)/2287)-1 mod n = 3911272796, which is a unit, inverse 494659176.

(2287 * 4523) divides n-1.

(2287 * 4523)^2 > n.

n is prime by Pocklington's theorem.