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.