Primality proof for n = 388425074903852603481408727235725774844022299:

Take b = 2.

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

468848762208042289 is prime.
b^((n-1)/468848762208042289)-1 mod n = 70743929769114021430385101625331568802521558, which is a unit, inverse 376277893427562830993133919023416860269232141.

16018527589 is prime.
b^((n-1)/16018527589)-1 mod n = 31696373501837656170653683103350101835274592, which is a unit, inverse 386361919353251368039360203084396896415184285.

(16018527589 * 468848762208042289) divides n-1.

(16018527589 * 468848762208042289)^2 > n.

n is prime by Pocklington's theorem.