Primality proof for n = 1135613:

Take b = 2.

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

1471 is prime.
b^((n-1)/1471)-1 mod n = 741221, which is a unit, inverse 1043786.

(1471) divides n-1.

(1471)^2 > n.

n is prime by Pocklington's theorem.