Primality proof for n = 22149492674086928081353:

Take b = 2.

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

173378833005251801 is prime.
b^((n-1)/173378833005251801)-1 mod n = 5525412857823478124482, which is a unit, inverse 8128672043349740015484.

(173378833005251801) divides n-1.

(173378833005251801)^2 > n.

n is prime by Pocklington's theorem.