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.