Primality proof for n = 5621767:
Take b = 2.
b^(n-1) mod n = 1.
32309 is prime. b^((n-1)/32309)-1 mod n = 2927843, which is a unit, inverse 3011739.
(32309) divides n-1.
(32309)^2 > n.
n is prime by Pocklington's theorem.