Primality proof for n = 38557:

Take b = 2.

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

17 is prime.
b^((n-1)/17)-1 mod n = 19123, which is a unit, inverse 16489.

7 is prime.
b^((n-1)/7)-1 mod n = 12907, which is a unit, inverse 18103.

3 is prime.
b^((n-1)/3)-1 mod n = 7606, which is a unit, inverse 10316.

(3^4 * 7 * 17) divides n-1.

(3^4 * 7 * 17)^2 > n.

n is prime by Pocklington's theorem.