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.