Primality proof for n = 11636921:

Take b = 2.

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

290923 is prime.
b^((n-1)/290923)-1 mod n = 8784011, which is a unit, inverse 7724096.

(290923) divides n-1.

(290923)^2 > n.

n is prime by Pocklington's theorem.