Primality proof for n = 222374623:

Take b = 2.

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

37062437 is prime.
b^((n-1)/37062437)-1 mod n = 63, which is a unit, inverse 134130725.

(37062437) divides n-1.

(37062437)^2 > n.

n is prime by Pocklington's theorem.