Primality proof for n = 6563:

Take b = 2.

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

193 is prime.
b^((n-1)/193)-1 mod n = 2528, which is a unit, inverse 4956.

(193) divides n-1.

(193)^2 > n.

n is prime by Pocklington's theorem.