Primality proof for n = 3938699322577:

Take b = 2.

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

222374623 is prime.
b^((n-1)/222374623)-1 mod n = 1505362245662, which is a unit, inverse 3427403832404.

(222374623) divides n-1.

(222374623)^2 > n.

n is prime by Pocklington's theorem.