Primality proof for n = 22561162540501040539:

Take b = 2.

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

5171003929967 is prime.
b^((n-1)/5171003929967)-1 mod n = 1881923577665948217, which is a unit, inverse 10911148973215549673.

(5171003929967) divides n-1.

(5171003929967)^2 > n.

n is prime by Pocklington's theorem.