Primality proof for n = 514313:

Take b = 2.

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

1213 is prime.
b^((n-1)/1213)-1 mod n = 227430, which is a unit, inverse 303624.

(1213) divides n-1.

(1213)^2 > n.

n is prime by Pocklington's theorem.