Primality proof for n = 2277413:

Take b = 2.

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

7207 is prime.
b^((n-1)/7207)-1 mod n = 430139, which is a unit, inverse 2058169.

(7207) divides n-1.

(7207)^2 > n.

n is prime by Pocklington's theorem.