Primality proof for n = 361223:

Take b = 2.

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

1009 is prime.
b^((n-1)/1009)-1 mod n = 344572, which is a unit, inverse 146997.

(1009) divides n-1.

(1009)^2 > n.

n is prime by Pocklington's theorem.