Primality proof for n = 337973:

Take b = 2.

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

4447 is prime.
b^((n-1)/4447)-1 mod n = 88713, which is a unit, inverse 135017.

(4447) divides n-1.

(4447)^2 > n.

n is prime by Pocklington's theorem.