Primality proof for n = 9538204373:

Take b = 2.

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

142183 is prime.
b^((n-1)/142183)-1 mod n = 4470685906, which is a unit, inverse 7310929661.

(142183) divides n-1.

(142183)^2 > n.

n is prime by Pocklington's theorem.