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.