Primality proof for n = 37573380636081815018124278995574282493401057:

Take b = 2.

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

15867926547803050208571480396443 is prime.
b^((n-1)/15867926547803050208571480396443)-1 mod n = 28360458484290330459184554906762597357863012, which is a unit, inverse 10175893365203686752831115121411545550709865.

(15867926547803050208571480396443) divides n-1.

(15867926547803050208571480396443)^2 > n.

n is prime by Pocklington's theorem.