Primality proof for n = 22009001472962227:

Take b = 2.

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

128262069029 is prime.
b^((n-1)/128262069029)-1 mod n = 10683752564916910, which is a unit, inverse 390336835632271.

(128262069029) divides n-1.

(128262069029)^2 > n.

n is prime by Pocklington's theorem.