Primality proof for n = 220355754737793598754693970385959669739:

Take b = 2.

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

40024942543673 is prime.
b^((n-1)/40024942543673)-1 mod n = 59783355271040864061130466512529280068, which is a unit, inverse 5615548840263839127593212930024375029.

20492504424583 is prime.
b^((n-1)/20492504424583)-1 mod n = 32873292013651753297056891962573109830, which is a unit, inverse 9236730932795534987487440329170238324.

(20492504424583 * 40024942543673) divides n-1.

(20492504424583 * 40024942543673)^2 > n.

n is prime by Pocklington's theorem.