Primality proof for n = 397429:

Take b = 2.

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

33119 is prime.
b^((n-1)/33119)-1 mod n = 4095, which is a unit, inverse 287857.

(33119) divides n-1.

(33119)^2 > n.

n is prime by Pocklington's theorem.