Primality proof for n = 1114261:

Take b = 2.

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

379 is prime.
b^((n-1)/379)-1 mod n = 1056733, which is a unit, inverse 666856.

7 is prime.
b^((n-1)/7)-1 mod n = 448487, which is a unit, inverse 173390.

(7^2 * 379) divides n-1.

(7^2 * 379)^2 > n.

n is prime by Pocklington's theorem.