Primality proof for n = 363557:

Take b = 2.

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

937 is prime.
b^((n-1)/937)-1 mod n = 221580, which is a unit, inverse 160772.

(937) divides n-1.

(937)^2 > n.

n is prime by Pocklington's theorem.