Primality proof for n = 5617163:

Take b = 2.

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

4451 is prime.
b^((n-1)/4451)-1 mod n = 4463567, which is a unit, inverse 3990902.

(4451) divides n-1.

(4451)^2 > n.

n is prime by Pocklington's theorem.