Primality proof for n = 315461:

Take b = 2.

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

15773 is prime.
b^((n-1)/15773)-1 mod n = 102192, which is a unit, inverse 141416.

(15773) divides n-1.

(15773)^2 > n.

n is prime by Pocklington's theorem.